In reality, of course what you say is true and the fact that Athlon could previde a few extra hundreds of MHz in the clock frequency was not decisive.
Athlon had many improvements in microarchitecture in comparison with Pentium III, which ensured a much better performance even at equal clock frequency. For instance, Athlon was the first x86 CPU that was able to do both a floating-point multiplication and a floating-point addition in a single clock cycle. Pentium III, like all previous Intel Pentium CPUs required 2 clock cycles for this pair of operations.
This much better floating-point performance of Athlon vs. Intel contrasted with the previous generation, where AMD K6 had competitive integer performance with Intel, but its floating-point performance was well below that of the various Intel Pentium models (which had hurt its performance in some games).