Cool stuff:

Formatted according to chronological order:
Names:
Entering specific names at the start of the game has special results.
Flowey results in: 'I already CHOSE that name!'
Toriel results in: 'I think you should think of your own name, my child.'
Papyru results in: 'I'LL ALLOW IT!!!!' (playable)
Sans results in: 'Nope.'
Aaron results in:'Is this name correct?;)' (playable)
Temmie results in: 'hOI!' (playable)
Undyne results in: 'Get your OWN name!'
Alphys results in: 'D-don't do that.'
Mett/metta results in: 'OOOOH!!! ARE YOU PROMOTING MY BRAND?' (playable)
Asgore results in: 'You cannot.'
Asriel results in: '...'
Frisk results in: the game starting 'hard mode'.
Chara results in: 'The true name.'
For 'Gaster', see "secrets".

Flowey:
Throughout the game Flowey keeps a close eye on you: you can often see him ducking away after you turn around.
As a result, Flowey remembers everything. Every person you hurt or kill.

Papyrus:
That shed? That's for when he "captures" you.

Undyne:
Undyne does not befriend murderers.
You can point at Undyne during her 'date'.

Muffet:
Eating spiderpastry during the fight will cause her to stop the fight.
Buy her pastry and she won't even fight you.
If you bought something from the spiders in the ruins, Muffet will eventually get a message that you did and cease the fight.
On a pacifist run, if you have not bought anything, Muffet will still stop fighting after getting a message you did not hurt any spiders.

Mettaton:
Mettaton reacts to a ton of entries in the essay. It even has a unique response for "Toby Fox".

Asgore:
Eating Toriels butterscotch-cinnamon pie during the fight will cause Asgores stats to drop.
Asgore will remember how many times you have died to him.

Epilogue:
Seemingly every character has extra dialogue.
Some amalgams are back with their family.
Characters will recall what you answered throughout the game. Don't fuzz about telling Undyne anime is not real: the girl will MAKE it real!

Other:
You can create a pool of water by using the watercooler. A flower will have grown there in the epilogue.



New game+
Flowey:
Will remember what you did on your previous run(s).
Spared Toriel then spared her? He knows.

Sans:
Remembers. See 'secrets' for more.

Chara/[YOU]:
Chara will mock you for doing a double genocide run.
You get a special pacifist ending after doing a genocide run. (It's not good.)

Genocide run:
Toby does not take kindly to those who destroy their world. Uninstalling the game and removing related folders does nothing to relinguish your SOUL.