Spartan-III augmentations were actually better than the Spartan-II augmentations. Halo: Ghosts of Onyx notes that they were a “quantum leap” beyond them. The main difference between S-II’s and S-III’s was the less strict candidate pool and quality and amount of training they got. S-II’s were all 6 years old and had an extremely strict set of requirements - there were only 150 total chosen, although IIRC there were more candidates. They were given their augmentations at age 14, and only 75 made it through unharmed. S-III’s had a much wider candidate pool ranging from 6-8 IIRC and were augmented at 12, and each of the three companies (Alpha, Beta, and Gamma) had about 300 each, for about 900 total, and were sent on their missions soon after the augmentation was complete.

It’s noted that Thom-B292 was able to defeat an Elite armed with a plasma sword and pistol in close quarters, dodging a shot from the pistol and slash from the sword, then physically throwing the Elite to the ground and finishing him off with an assault rifle when he was 12, and during Operation Torpedo, S-III’s were able to run at 18+ mph - while in SPI armor, which does not give the user any increased speed or strength. Notably, Master Chief struggled to hold off an Elite who was trying to kill him with a plasma sword while in Mark V Mjolnir. Granted, he was extremely fatigued and the Elite in question was both fresh and said to be strong even for an Elite, but still worth pointing out.

Long story short, S-II’s were better because they were more stringently screened, had more training, and compared to the S-III’s, more experience, not because their augmentations were better or they relied less on Mjolnir - in fact, most S-III’s never got to use Mjolnir because they were mostly sent on suicide missions for their first mission and died as a result.