What does more amplifier power mean?

I've always been under the impression that more power actually improves sound quality/ sonic accuracy. When you compare similar styles of amps it just seems that drastic increases in power like the differences between average receiver amps that output about 100 watts per channel and a pro amp like the Behringer EP2500 that outputs 450 watts per channel results in huge differences not only in volume but the realisticness/depth of the sound you hear while only changing the amplifier and keeping everything else constant. What exactly happens when a speaker is fed more power? Is it my imagination or am I hearing something tangible that can be explained? It just seems that a piano sounds more like a piano. What does more amplifier power mean? Is it merely volume or is there something more?

No, volume is actually a function of gain, not power; you had it right.

There's a lot more to sound quality and sonic accuracy that just more power. The two attributes are not necessarily related to increased power unless the speakers are highly inefficient/insensitive and power-hungry, in which case the higher powered amp may sound superior. It is possible to have a poorly designed high powered amp that sounds horrible on a set of speakers versus a lower-powered amp that is very well designed and sounds wonderful on the same set of speakers. All else being equal, the higher powered amp is likely to sound more dynamic and effortless in a given set of circumstances -- which I suspect is what you have experienced thus far.

I think it comes down to "Headroom" or as I look at it, what the amp has in reserve, if
run at high volume. I use very power hungry vintage speakers that really "opened" up
when a 300wpc amp was added(Hafler XL600). That was a bit of overkill, I play Rock
music sometimes on the loud side, and the speakers became a bit screechy, so I'm
in the process of switching to something in the 200-250wpc range(SAE 2400L). High
power equals" Presence", with my speaker combo.
:14:

underpowering speakers can cause clipping when played loud,
but adding more power will increase the dynamic range, and depending on the quality of the watts, it will also improve detail and soundstaging...

but with your speakers,
and already having so much power,
it would be a waste of money to add even more power...

Rock on,
Bert.

I think it's really a question of balance....

Yes, if you have two amps of the same quality but one has more power, then it will likely sound better than the other as it is better able to control the speakers and less likely to clip...

However, Just dumping as much power as possible into your setup is not the best way to get great sound. If you have efficient speakers, then adding more power will produce far less impressive results, than replacing your existing amp with a higher quality amp with the same power output.

Try and balance power versus quality with the efficiency of your speakers, room size and your desired listening volume.