Why do speakers improve with more powerful amps?

I remember when I had my Adcom setup GSA 555-2 (250 wpc).  They were bridgable mono's, so I purchased another one of the same amps and series.  The end result?  My system sounded like a PA system.  Loud as hell and very unpleasant (sibilant).  I still like a lot of power because I like to feel the orchestra or the rock band playing, but now it's with tubes.

@ghdprentice 

Very nice main system and the space, lots of audio research very Tubi. The room looks dead quiet. Another world.

Also like the Tiffany lamp, and really love the doggie.

Use of the meridian processor and speakers for Home theater is interesting, not many of these systems are out there.  I like the idea of separating the HDMI signal which I believe is how it works. Wish more systems would do that sort of thing. 

I'd love to answer but there are so many variables it would take a small book. But all things being equal my friend George Bischoff who built triode tube amps(Melos) with over 400 watts/channel always said a good big amp will beat a good small amp every time. Notice the word good which encompasses many things. But it will always include a superior power supply which is costly and can be compromised without affecting standard specs. 

"headroom" is also a marketing term.

Utter garbage! Don't make wild, unsubstantiated assumptions. If you don't understand the term, it's not that hard to find an explanation. You don't even need a book. Just a computer and browser, so there can be no excuse.

Correct questions should be "why some speakers need more powerful amps whereas some other speakers don't?" Because choice of a speaker is a highly personal one like finding your spouse. Once you find the speaker of your choice, next you try to assemble rest of the equipment including the amplifiers to get the "best possible" sound from it.

Two factors determine whether you need a powerful amp or not: Internal impedance and sensitivity of the speaker. If you have highly efficient speakers >90dB, then you don't need a powerful amp. If you have an inefficient speaker <90dB with impedance dropping below 2 or 3 ohms, then you will need an amplifier that can supply current at these loads, i.e., more powerful amp.

There are many variations in between these two scenarios, but you get the idea. The nominal impedance 8 ohm or 4 ohm does not tell you the whole story. Internal impedance change with frequency and can be resistive only at resonance frequency, but rest of the time it is either capacitive or inductive. Some speakers have a nominal impedance of 4 ohms, but minimum impedance at certain frequencies can drop below 3 ohms or even  2 ohms! So amplifier need to provide sufficient currents to these inductive/capacitive loads and at the same time should provide enough dynamic currents for the music.

As you can see, we can only decide what works best by listening. In the end, only you can decide what is best for you. These are just guidelines so that one can navigate and resolve any inadequacy in the audio chain.