Note, all comparisons are made against a berserker dual classing at level 9. 13 is too late and 7 sacrifices too much, additionally a level 9 berserker to mage will have 250k in each class when it gets its fighter abilities back, and can be done immediately upon leaving irenicus dungeon, making it the clear patrician choice. Additionally, kensai dual is inferior to berserker, so I will not bother comparing the F/I to kensaige.
Additional note: I got bored as hell writing this, so forgive me if it becomes boring or unreadable.

Let's break everything down down, one by one.
We'll talk about the fighter half first then move on to the illusionist
Thac0
What it looks like on paper : 

BM thac0, 11
F/I thac0 0

How it actually is :
BM superior in bg1 by a landslide
BM superior in bg2 by a slightly smaller landslide, tied at the end
BM and F/I are both not going to miss in ToB


Fighter/Mage has equivalent thaco at level 14/14 at 3 mil xp, then it starts to get better than the berserker dual, until this point, the berserker has better thac0 because of grandmastery and berserking, which give +2 and +2 respectively OVER what a specialized F/I gets. Thac0 matters most in bg1, somewhat in bg2, and is irrelevant in ToB.

Why is it irrelevant the later you get in the game? In bg1 you will make a char with 18 str, boost to 19 with tome, 21 after hell, and 22 after lum the mad (alternatively you can use the belt of fire giant str for the exact same thing). This, combined with the fact that the highest base AC in the game is -12, +3 or higher weaponry means you're sitting on a thac0 that is better than 99% of enemies AC, meaning you never miss outside of critical misses.

Base thac0 + grandmastery +3, +5 from your weapon, +4 (or more) from your strength, +X from equipment (balduran helm, etc) + 2 from berserking, + Y from additional buffs from party members +Z from debuffs = you landing every attack on anything that isn't demogorgan, the ravager, Amelyssan, a drow in full plate (Depending on your weapon of choice) or a dragon, if you consume/buff (which you should) you won't miss these either. Let's go over each game.

Bg1 thac0 : Berserker superior for 100% of the game, no contest here, high mastery + berserking + pure fighter progression guarantees the F/I is significantly behind in thac0 when it matters the most. Advantage: berserker

Why does thac0 matter the most in bg1? AC scales better than thac0 here. Full plate +1 is the best armor a non mage can get, in bg2 (aside from drow equipment and big metal unit) full plate +3 is the BEST AC armor a fighter can get (Aside from drow full plate, which is only worn by a small handful of fighters). That means despite thac0 more than DOUBLING for a pure fighter in bg2, the chestpiece (highest AC contribution) only increases by TWO. What about caster AC? Mages ALSO get their ONLY armor spells in bg1, spirit armor, imp invis, blur, this means aside from a minor bonus from equipment in bg2, mage AC doesn't change much either. What does this mean for the berserker? This means the thac0 he has over the F/I early on is VERY useful, and the thac0 the F/I has later is literally USELESS.

Bg2 thac0 : Berserker superior for 90% of the game, at 3.5 mil xp the F/I pulls ahead by 1, at this point thac0 starts to become inconsequential (if not already), advantage : berserker.

At this point the berserker has grand mastery, giving +2 thac0 combined with +2 from berserking giving him a +4 thac0 bonus over the F/I, this allows the berserker to dual class to a mage and STILL have better thac0 until the F/I hits 14 in the fighter class, which is at 3 million xp, late in SoA. At this point they're equal, and as time progresses the F/I will slowly pull ahead, but it's so late in SoA that it's ultimately irrelevant. During SoA, the highest AC enemies will be drow in full plate, this is AC at its zenith, from this point on, AC will remain the same or be worse, when you reach this point, the berserker is still ahead of the F/I in terms of thac0. As such, the berserker is the clear victor here, having higher thac0 for almost the entire game, and being on PAR for the VERY end.

ToB thac0 : Fighter/Illusionist has superior thac0 for 100% of game but its now irrelevant, advantage:none

As mentioned above, at this stage in the game there are no enemies in the game that you are going to miss on, AC will never surpass what it was for drow in full plate and in 90% of fights enemy AC will be less than the drow you encountered in the underdark, the exception to this obviously being drow wearing full plate in sendais enclave, dragons, demogorgan, the ravager, and amelyssan (Depending on your mods, some demons will have high AC as well but at this point its not enough). You already have enough thac0 to hit all these enemies, as such, the F/I bonus to thac0 here is irrelevant. Lastly, if you're running with mods to make enemies have better AC than also consider BM can timestop auto, the FI cant do this until late ToB.


Hit points

No need to link anything here, obviously a pure fighter dual classing at 9 will gain full advantage of his d10 hp per level rolls and +5 hp from constitution. The F/I will be down 23 hp at 64k and 40 at 161k. Once again, hp matters most in bg1 than in later games, as immunities such as stoneskin, pfmw, pfme, etc, make hp far less valuable than it was early on.

Bg1, Bg2, ToB : Berserker superior 100%, although this matters less in bg2 and ToB.

Attacks per Round
BM 1(base) + 1(Grandmastery) + 1/2(level 7 fighter) = 3 APR (game rounds)
F/I 1(base) + 1/2(Specialized) + 1/2 (level 7 fighter) + 1/2(level 13 fighter) = 3 APR (game rounds)

On paper they look even right? Berserker gets the same attacks per round as a F/I because he will get grandmastery, while the F/I gets the same bonus at 13/13. Wrong, if you think about WHEN you get these bonuses, once again berserker is superior.

Berserker gets 1/2 at 64k (level 7 fighter bonus), 1/2 at 250k (grandmastery)
F/I gets 1/2 at 128k (level 7 fighter bonus), 1/2 at 2.5MILLION (level 13 fighter bonus)

Just for comparison sake, the F/I needs to have TEN TIMES as much xp as the berserker to have the SAME attacks per round in SoA. Crazy right?

As we can see here, berserker gets its APR faster than a F/I. Not only this, but oils of speed round up APR in bg1, so a berserker will have access to 4 APR while dual wielding at 64k with oils while the F/I only gets 3 until he reaches 128k xp, late in BG1. In bg2, grandmastery will be achieved before leaving the promenade, while the F/I will be in spellhold or later before seeing that sweet 1/2 attack bonus. In both games, berserker gets a clear advantage in APR, especially when combined with oil of speed and improved haste (which ofc turns 1/2 into 1)

Physical damage dealt

Grandmastery gives +3(its +5 but i subtract 2 because FI gets 2 damage bonus via specialized and I want to make a fair comparison) per hit to the berserker, berserking gives +2 for a bonus of 5, this, combined with the fact that the berserker has superior thac0 means he will be putting out more physical damage than the F/I at ALL stages of the game.

Wait a sec, what about the dual class transition? Aside from the fact this is a very short amount of time, if you keep yoshi you can steal scrolls and unlock your kit before even leaving the promenade. If you don't want to steal you can do the circus as a level 1 mage with the wands from irenicus dungeon, implicate the tanner, turn in the acorns, talk to madulf, and pick up a scroll in merellas cabin. All of these quests require ZERO combat (aside from killing two orcs and kalah in the circus) and will give you enough xp to get your dual class back without learning a single spell.

Hold on, what about fighter HLAs? 

You shouldn't be using them, everytime you pop a HLA it uses your action for the round, meaning you can't cast a spell, and there are almost zero times where using a warrior HLA is better than casting a spell. Let's go through them one by one.

Hardiness: You have stoneskin, mirror image, armor buffs, and pfmW, you are not taking physical damage unless you aren't paying attention or go afk.
Warcry: Useless, a level 2 horror spell as an HLA is embarrassing
Deathblow and greater deathblow: Useless, wasting a slot here is almost as bad as warcry, enemies this would be great for are too high level and enemies that this works on will die in two seconds anyways.
Resist magic: You got protection from fire and magical energy, magical damage is not an issue, and with berserking and save bonuses from spells and equipment you are not going to get cc'd or take serious magical damage. 
Smite: Can only be taken once and will push your enemy away from you so you can't even attack it with melee, cool ability for an archer, not all that helpful for a F/M
Whirlwind and critical strike: This is where again people aren't thinking, on paper, these look great, but if you break it down, they turn to shit fast.

Whirlwind

You are already at the APR cap with improved haste in essence as a BM, you ARE if you use belm in OH, and because you're a mage, it's not going to be dispelled. WW does nothing for you here unless you're using a 2h. Let's compare shall we?

We will assume both the F/I and the BM have the same gear, so we can remove the damage bonuses from their strength and equipment.
Grandmastery + berserking = +5 damage bonus OVER the F/I (Grandmaster +5 - specialized +2 = +3, which combines with the +2 from berserking to make a +5 damage bonus per swing)

Let's say you're using an axe on BM, and a 1d12 2h (best case scenario, most 2h are 1d10 not 1d12)

1d8+5 x 9APR = 36 weapon damage + 45 berserking/mastery bonus = on average 81 damage per round
1d12 x 10APR = on average 60 damage per round for using GWW

If the berserker is using a katana it makes it 90 to 60 average damage, and if the 2h wep is 1d10 its reduced to 50 for the F/I using whirlwind. No need to add strength or item boosts as these are the exact same between the two.

What does this tell us? It tells us that a BM is doing more damage per round JUST AUTO ATTACKING than a F/I using a two handed weapon WITH GREATER WHIRLWIND.

Even if we pretend that you're using the +4 2h in bg2, or the +6 2h in ToB, it's only going to be one damage higher than a +3 or +5 weapon used by the BM. Which means you'll STILL be doing less damage per round. Not only are you doing less damage per round than a BM, you're popping a HIGH LEVEL ABILITY to do it. 

What about the ravagers decapitation effect? If you wanna go all the way to the bottom of watchers keep be my guest, and while you're down there you can pick up the axe of the unyielding which does the exact same shit (albeit with a -4 penalty save), and funnily enough, you get it before the ravager as well (it does seem to be the case the berserker gets everything first, and better than the F/I doesn't it?)

HEY THE RAVAGER DOES DAMAGE TOO

True, 3-18 bonus damage per swing on a failed save, averaging an extra 10 damage per failed save so we'll say an extra 50 damage per round (assuming they fail their saves half the time, generous to be sure, even with malison and doom up they will save a lot more than this)

Now, this looks better right? You're now doing 20 damage per round more than a bm. But since you just popped an HLA, you can't cast a spell. That means as a BM if I cast any damage spell at all, even one as simple as magic missile, I just did more damage than you. A BM casting a level 3 spell like skull trap (illusionists don't get this) or flame arrow will do FAR more damage than a F/I using greater whirlwind every round. The point of being a fighter mage is not to just cast defensive spells, or swing your sword and use magic sometimes. You should be casting a spell every round then beating the snot out of your enemies. Casting ANY sort of damage spell ESPECIALLY AoE will do WAY more damage than an HLA. If you're wearing robe of vecna(lets be real, who isn't) you can add literally HUNDREDS of extra points of damage each round to your rotation, WW barely does more than a magic missile. It's not even close. WW is now officially garbage for a F/M, but what about critical strike?

Critical strike will guarantee you land every attack and it will do double damage to anything that isn't immune to crits.

This is by far the best thing a F/I gets, and it's very close, but still inferior.

Once again, when you pop this ability, you can't cast a spell that round. It is almost ALWAYS better to weave a spell into your auto attacks than critical strike. Whether that spell is a horrid wilting/skull trap, a sequencer, a defensive spell (PfMW), or an offensive spell (Breach, ruby ray, etc). This is the real core of the problem with fighter HLAS as a F/M.

If you are a fighting a wizard, he's going to have protections on that need to be removed, making offensive spells better than an HLA
If you are getting hit a lot, it is better to cast defensive spells to prevent you from being killed.
If you are fighting more than 2 enemies, it is better to cast an aoe spell (horrid wilting, incen cloud, skull trap, comet, Dbreath) than use critical strike or whirlwind.
If you are fighting one enemy, and it's not a wizard, and it's not attacking you quickly, and it doesn't have protections up that need removing, and it doesn't have high physical damage resistance, then it is in this case better to use critical strike than nothing at all. The cons of F/I do not make up for a "win more ability" which is basically something that is really irrelevant in the fight, but nice to have. You have already won the fight if you're are fighting somebody with the above requirements, so killing him slightly faster doesn't really help much. You cannot critical strike an enemy that is invisible, you need to cast a spell for that, you cannot critical strike an enemy with pfmw, you need to cast a spell for that, you SHOULD not critical strike an enemy that has stoneskin, you should cast a spell for that, you SHOULD not critical strike an enemy that just broke through your stoneskin and is now mauling you, you SHOULD cast a spell for that. Everytime you cast a spell or use an HLA you can't use the other for that round. 99% of the time, you are going to cast a spell, not use an HLA, for this reason, WW and critical strike, which look great on paper, are next to useless. If they lasted more than one round (like berserking for example) they would be fantastic, but they don't. If you could pre pop them before a fight they would be amazing, but you can't. In almost every situation in this game, in almost every single fight, it is better for you to cast a spell than use an HLA, this is why they are bad for fighter mages.

The power of a fighter mage be it dual or multi stems from the fact that physical damage is the highest sustained dps in the game, combined with arcane magic keeping them immune to pretty much everything, and at every chance you get, breaking down immunities or hitting them with raw spell damage. Critical strike, WW, hardiness, MR, they do not fit here. The ones that increase your defenses are superfluous and inferior to arcane magic, and the damage HLAs are going to be inferior to magic 99% of the time.

A sequencer of skull traps (which you can trigger, as well) will do several hundred damage to an aoe pack, a flame arrow will do massive damage to a target that is not immune to fire, a horrid wilting / cloud / comet / dbreath will do more than critical strike ever could to even a small group of enemies. The only possible argument you could make for these abilities is that they MAY do more than an illusionist normally can (the illusionist does lose skull trap/ horrid wilting after all, not to mention it will always be underleveled compared to a BM at the start of bg2 all the way to the end of the game) but that doesn't make up for the fact that a BM does it better. You can argue that a F/I has tools that compensate for what an illusionist loses, you CANNOT argue that they are superior to a berserker mage, because they aren't.

So far we have established the berserker mage is superior in thac0, physical damage dealt, HP, and that the fighter HLAS are useless at worst or superfluous at best. Now lets talk about saves and spells

DEM SAVES 

F/I Gnome 18 con level 1 Compared to Berserker level 1
14 	14 Death
6 	16 wand
13	 15 poly
15 	17 breath
7 	17 spell

F/I gnome 18 con 250k xp compared to BM (dual classed at 9, the patrician choice)
10	8 death
4	10 wand
11	9 poly
12	9 breath
5	11 spell

F/I Gnome 18 con vs bm 1mil xp
8	8 death 
2	7 wand
9	9 poly 
9	9 breath
3	8 spell

F/I Gnome 18 con vs bm 2 mil xp
7 	8 death 
2	7 wand 
8	9 poly 
8	9 breath 
3	8 spell

F/I Gnome 18 con vs bm 3 mil xp
5	8 death
2	5 wand 
6	7 poly 
5	9 breath 
3	6 spell

Max level F/I gnome with 18 con vs bm
3 	8 death
0 	3 wand
4 	5 poly 
4 	5 breath
1 	4 spell

(all comparisons made with berserker at 250k mage at the XP mark -250k from dual class)

From this we see the FI has a worse save vs death and poison early on and superior saves across the board, especially spell until late game. This means the F/I is 10x better right? Only on paper, let me explain.
Save vs wand is next to useless, even with scs there are less than 5 npcs that use wands in this game, without scs there is maybe 1 or 2. That goes for the entire trilogy. So save vs wand is useless.

Poly and petri apply to 3 spells/abilities. Basalisk petrification (irrelevant for obvious reasons) polymorph other (extremely rare, I can't think of a time this has been cast on me with or without SCS) and the gaze effect of aeletec and some demons(again, irrelevant because of consumes). Flesh to stone and sphere of chaos for whatever reason requires save vs spells, not petrification. This makes Save vs poly/petri ALSO useless

Breath saves are only used against dragons and a few more depending on mods, wing buffet, which give a WHOPPING -10 penalty to save against, is something that berserkers are actually immune to (trademeet belt should be on a fighter, not a wizard, so don't try to use this as an excuse) If you are not using spell immunity or protection spells vs dragons, you're doing it wrong. This in short, makes save vs breath utterly useless as well.
Save vs death/poison pops up ALL the time, in a variety of spells and monster abilities including undead and spiders. Ironically this is the one that berserker has BETTER than the F/I. So advantage here to berserker.

Most importantly of course is the final one save vs spell. F/I has this in spades early, but the bersker also has TWICE the hp of a F/I in addition to having immunity to most CC effects. As such the advantage in bg1 is rendered moot. You're immune to cc and may sometimes fail your save vs spell against a fireball but who cares, you got twice the hp of a F/I so taking double damage isn't a big deal, worst case scenario you can just pot for the ONE or TWO fights in the game where this could become an issue.

Even then the fact remains that this is unbuffed. When you add blur, Imp invis, and spirit armor your save vs spell is reduced by 8. (F/I miss out on this awesome spell). This is BEFORE EQUIPMENT OR OTHER BUFFS. When you add that too, suddenly it doesn't matter if you're a gnome or a human, your saves are good enough to save vs anything in the game at LEVEL 9. 
Saves are very important, but it's so damn easy as a wizard to cap them out it doesn't really matter what race/dual/multi you are.
Once again, the F/I looks great on paper, but in practice the BM is not only damn fine in this area, but in some ways better. Getting dispelled happens all the damn time in bg1, so the gnome wont have those bonuses from blur/invis etc, if he's hit with a malison / doom, he's gonna fail his saves, and then his awesome saves are USELESS while the berserker doesn't give a fuck because hes angry, he doesn't need to save vs spells since he can't be cc'd. 

Advantage here, slight edge to berserker.

Finally spell progression, obviously in bg1 F/I has spells, BM doesn't. But who cares, damage spells scale poorly, and the good spells the F/I can't cast until the end of the game (stoneskin) or cant at all (chaos). He has to read them via scroll or use wand (which granted is still good). So, advantage here is obviously to the illusionist, who also can summon a familiar and double dip to get a little extra hp going into bg2. (Your familiar is deleted via irenicus dungeon, but the hp it gives stay)
So now we're at 250k for BM and F/I. We both have level 5 spells but the F/I has 1 more of each via specialization. At this point the F/I is still the superior spell chucker, but that changes fast.
At 1mil total xp the BM has level 6 spells, (imp haste) the FI wont get this until 1.5 mil
At 1.75m xp the BM has level 7 spells (Project image, ruby ray), the FI wont get these until 3 mil xp
At 2.5mil xp the BM has level 8 spells (horrid wilting, Incen cloud), the FI wont get these until 4.5 mil xp
At 3.25mil xp the BM has level 9 AND 10 spells (Time stop, dark planetar, Imp alacrity), the FI wont get these until 6 mil xp.

Additionally, every spell that scales with level will be cast better by the dual class, their spells will last longer, with more charges, and do more damage. Every single one because they're not wasting xp in a useless class. The biggest drawback of the FI is delaying HLAs (mage obv, fighter hlas are crap). A planetar can solo pretty much anything, even with SCS. Timestop also means your opponents cant fight back, nor can you miss, even against dragons. The BM gets all of these WAY sooner than the F/I does. If you play solo this isn't a big deal as you absorb all the xp yourself, but if you're in a party this is a serious problem. 6 mil xp depending on your playstyle is mid to late ToB, you're basically getting timestop at the point where enemies start to become immune to it. What a joke.

Illusionist bonus spell slots? By the time they reach the required level to unlock the same spells that the BM had ages ago, the BM has already gotten more levels which unlock more slots to be on par(or have more) than the damn F/I. I don't get why people rave about the FI it's literally worse in every damn way.

You're sacrificing HP, Thac0, damage, spell progression, spell power, all necromancy spells, and putting off your most potent abilities until the very end of the game just so you can grow a beard? It's not even a GOOD beard. If you could be a dwarf fighter mage you might have been able to convince me to look past just how SHITTY it is compared to a BM, but not a gnome.

FYI: the F/I is still one of the best classes in the game, it's just not nearly as good as BM. That doesn't mean you shouldn't play one either, if you like playing F/I than by all means make one, this is just an argument to show why they're not as good as the BM. There's a ton of misinformation that has been spread for DECADES about this game (monk late game powerhouse lul), and I'd like to set the record straight.

Finally, if you're still somehow reading this, if I missed something let me know. God bless.