The power supply of solid-state amps is critical. With almost all opamp-driven amps being essentially the same, the power supply is generally the only difference and I'm sure you know how different opamp-based amps can sound, even if they use the same opamps.
For transistor/MOSFET amps, there is still a large difference in PSUs and they do make quite a difference. On an amp like the PPAv2, I have ran it off a slightly chopped up version of a TREAD, a complete TREAD and off STEPS. There is definitely a difference.
Of what I actually "hear" with higher end PSUs, there is less noise, improved clarity and better bass impact.
On top of the PSU itself and of course components used, the transformer choice is also quite important as a cheap transformer will add a lot of noise into the circuit, like A LOT! I've built an amp that sounds like it has a small car engine inside of it because I used a junk transformer, even though it was toroidal; replaced with a nicer Amveco model and 90% of the noise went away.