basically when you tell an audiophile to talk about amps, he will answer "power". most people don't understand amp designs and don't understand basic electricity like the interactions with a given load. in short they don't have a clue and you shouldn't even care about what they say.
now those same people always have an opinion on which sounded better and that's natural. even I have personal taste. but if they simply said "I preferred this one but I don't know why", then they wouldn't look like expert audiophiles now would they?
ask how they came to their opinion, and most likely you'll know why you shouldn't care about their opinion. the method is as important as the conclusion and most audiophiles have no method.
of course loud isn't everything, for example there is a reason why we get amps max power @1%THD into a given load instead of just being fine with max power. thinking that how loud it goes is all we need, is the same nonsense as thinking we can explain how something sounds with frequency response alone. it's a great information, but sound is still more than 1 variable. knowing that we go loud enough or that we get enough power(which is correlated once you're set on a given headphone) will help know that the system can work as intended, but it doesn't tell us about fidelity or taste.
for power, if the load(headphone) is known, and the loudness is known, then you have a pretty good idea of the power used. for those not clear about electrical behaviors with a load in series, think about the almost ok water analogy for current and voltage.
the pipe is the headphone, the amp is the water tank. a really bad amp is when the pipe isn't the limiting factor(bad impedance ratio, low current...). most modern headphones and amps should work in an "impedance bridging" system where the diameter of the pipe should be way smaller than the diameter of the water tank. that is to say the impedance of the headphone should be way bigger than the impedance of the amp. once that is assured, it's pretty obvious that the current flow will be strongly limited by the headphone and not by the amp, the impedance is more or less stable, so what's left to care about is voltage/loudness.
obviously if you start driving a 50ohm headphone with a 120ohm amp, then it's a whole other game as you're now using a pipe larger than the water tank ^_^. but when you do what you're supposed to do and keep a good damping ratio, then enough voltage is often the same thing as enough power as long as you don't get into clipping or mad distortions(thus the @1%THD for power into a load being a nice extra information).
different loads will have different needs, and some amp designs will not behave as well with all loads. but there are reasons for that, and just saying it needs more power to look like we know stuff is like saying I eat baguette because I'm french. it looks legit, and in fact often enough it will be true, but obviously sometimes it was a croissant and you look like a fool when you call my croissant a baguette.
now if you'll excuse me, I have my meeting with the Analogy Anonymous. I want to stop using it's just so hard!!!!