People also say that my (former) magnepans, which are at least a bit harder to drive, generally, only sound good with lots of power. They were nuts. The speaker gets however loud the incoming voltage tells it to get. More voltage = louder. Also = more power. Conversely, less loud (like during silences or when people are whispering, = even less power. SO a lot depends on how loudly you listen. Technically what happens when you run out of power is the amp starts to distort, which will, oddly enough, make it sound more "full" at low distortion levels, and may even give the impression of more bass. If distortion isn't audible with your current power, more power will only serve (assuming the amp is otherwise accurate and not unduly noisy) to make the inaudible distortion even more inaudibler. Meaning no real difference to you.
Now. I've owned more and less powerful standalone amps, and certainly would not tell you that your world is going to end, or the sun is going to go supernova if you acquire additional power that you don't need. It might even improve your listening experience simply by removing a potential source of worry. But power per se, if not used, won't make a real difference.
Numbers of folk will jump on here to espouse the view I used to hold, that somehow the unused headroom changes what the speaker does, as if the speaker has a "potential power sensing" organ and can just tell that, if it needed it, amp B could supply more power than amp A. Ohm's law dictates that the power delivered at any given voltage gain is exactly the same, assuming sufficiently low levels of distortion and otherwise appropriate amplifier design. And you can certainly ride that train if you want to.
IMO, if people really hear a difference between amps based purely on power it's either because the lesser powered amp has some level of detectable distortion, even if you don't think it's sounding distorted, or because (this is kind of a different way of saying the same thing) they can play higher levels, cleaner. Then, it's the higher level that sounds different, because (a) loud sounds different (simplest example being equal loudness curves making the bass more louder) and (b) once you push the speaker hard enough it will compress dynamically, which people might find pleasant even though it's inaccurate, and/or the speaker may distort quite audibly (10 times higher distortion in speakers than amps is pretty much par for the course) and the speakers' distortion may sound pretty good, too, even though it is inaccurate. Certainly guitar players the world over have enjoyed the nice sounding combination of amplifier distortion plus speaker distortion since before rock and roll was invented . . . so say the 1940s or maybe a bit before . . . .
FWIW, if I had a ton of money I'd likely get Bryston amps, just because they measure astoundingly. But they're relatively expensive . . . I did love my Bryston, though, for reasons I suspect were more psychological than aural given my relatively low listening levels (my peaks probably don't consume 20 watts per channel, so big power is not really a big deal to me). Even a little louder (3 dB) pushes you to possibly 40 watts per channel, which can start to be challenging for some receivers when everything's really cooking on peaks (these last way way less than half a second in normal movies, but they might occur often). And with my system listening at reference level (a) is impossible because my speakers will be compressing at those levels and (b) might legitimately call for over 100 watts per channel, perhaps as much as 200 (that's 10 dB louder, about twice as loud compared to my maybe-20 watt peaks . . . .