Power cannot affect sound, no. More power equals more volume. Discussion of transients and such isn’t really a thing for headphones, especially at the frequencies reproduced.

That said, if an amp is being pushed to its limits, there are definitely some that begin to distort as you get close. So an amp with lots of headroom to avoid distortion is going to be very helpful.

Unless you can tell them apart in blind, level-matched tests, the differences could extremely easily be imagined due to expectation bias, or due to slight differences in volume. And no, you cannot use your ear to level-match, it’s not precise enough, you need a multi-meter.

I’m not sure why everyone seems to believe that their hearing is so accurate, it’s really not. It’s almost entirely in your brain, making sense of vibrations, but what you EXPECT to hear, or what you THINK you’re hearing changes what you actually hear. All the time.

The McGurk effect is an obvious one, but your entire range of hearing works this way. Human hearing is incredibly unreliable and fallible, this is irrefutable fact. Perceiving a difference doesn’t mean there actually is one.

Can you explain why those differences go away entirely when you blind and level match the testing?

I’m a recording engineer, and I’ve tested dozens of amps and DACs properly with several people who definitely claimed a night and day difference between source gear, yet couldn’t even get a 60% accuracy rate on correctly telling them apart.

They only heard the differences when they knew what they were listening to. I’ve done this several times over the years, and my findings line up with the actual research done.

Frankly, you perceiving a difference just listening isn’t really proof of anything do to how incredibly easily influenced human hearing is, it’s incredibly fallible.