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.
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.