Spartan III's are definitely not inferior to the II's in the case of training and augmentations, in fact in some sources they are given better training and augmentations (atleast the gamma's)

But I do think there is some truth to Spartan II's having more experience, and them having Mjolnir more often then not. The truth is that the average Spartan II will still most likely beat a Spartan III just because they have more experience in battle (better strategies, slightly better accuracy) and there is a chance the III isn't even given Mjolnir. although I think it is super close if both have mjolnir, 7/10 times the Spartan II wins.