The SPARTAN-IIs underwent more intense surgical augmentations than the SPARTAN-IVs did, and there was also the fact that every single one of them were indoctrinated and mentally conditioned from eleven years old to be the perfect soldiers. In terms of the hardware that they used while they were still in active service prior to the SPARTAN-III program, the MJOLNIR Mk. IV-VI suits were still not as advanced as the MJOLNIR GEN-2 suits that the S-IVs used, but their sheer efficiency and their more intense augmentations give them an edge that the S-IVs sorely lack.

There is also the fact that S-IVs were not genetically-screened child abductees or war orphans raised from childhood to be relentless killing machines, but consisted of volunteers from consenting adults, ranging from promising recruits fresh out of Basic to highly decorated veteran soldiers from other branches of service (e.g. Marines, ODSTs, etc.). This lack of mental conditioning and experience compared to the SPARTAN-IIs, as well as the less intense augmentations, is made up for by their more advanced MJOLNIR GEN-2 suits, but ultimately, this more advanced armor is merely a crutch. Outside of their MJOLNIR suits, an S-II would handily trounce an S-IV. Even if they both used the respective hardware that they are known for, the odds would still be stacked in the S-II's direction.

It's often said that an experienced soldier with only the basic equipment will outperform a newbie with the best gear, and the S-IIs demonstrate this very clearly. The S-IVs underperformed in multiple regards during the Requiem campaign, and while they were significantly outperforming regular Marine and ODST detachments, they were still falling short of the incredible wonder-soldier legend that the SPARTAN-IIs had woven for themselves. The Covenant Remnant still managed to outpace, outthink and outmaneuver the SPARTAN-IVs at numerous turns, while the Covenant during the course of the Human-Covenant war NEVER saw the original SPARTAN-IIs coming.

The S-IIs were so horribly efficient at waging both physical and psychological warfare against the Covenant that they soon came to be called 'Demons' by the Covenant - faceless, nigh unkillable slaughter machines that would appear from the shadows, slaughtering everything in sight, and then fade without so much as a trace. Elites would be wary of them, Brute bravado would cease to a halt, and Grunts and Jackals would pretty much just soil their pants at the mere thought that a 'Demon' was nearby. The legend of how 'Spartans Never Die' was woven through how they were just so damn hard to kill, that even if you did manage to take one down, it would usually be at the expense of an entire armada of your own troops.

Compare that track record to the Requiem campaign, where the SPARTAN-IVs were merely augmented marines and ODSTs wearing really advanced armor, and they sustained multiple public casualties that were highly visible to their enemies while being outsmarted and outplayed by their foe multiple times. You can see the kind of damage the Spartan legend would take after that kind of performance, and the Covenant would begin to see the Spartans not as demons, but as simply very fast, very tough humans that can still be killed.

From a storytelling standpoint, it's easily explained - Conservation of Ninjutsu. The S-IIs were a very small, select group that could not introduce any new members whatsoever, while the S-IVs are practically an entire corp unto themselves, with more recruits joining every day. It's not hard to see which group would outperform which if there was only a finite amount of 'ninjutsu' to go around between each Spartan, making the S-IVs much less effective than the S-IIs.

Long story short, yes, a SPARTAN-II would be significantly outperform a SPARTAN-IV, despite the latter's superior technology and better gear, simply because the SPARTAN-IIs have the edge in genes, training and mindset.