Why do speakers improve with more powerful amps?

The answer is very common: " It depends. " An inefficient speaker with a low impedance is not going to work very well with a 2W SET amp. Thats a fact. Paired properly, an amp/ speaker combo CAN sound wonderful with just a few watts. It all comes down to proper matching.... There is certainly a need for power in many applications.

Two points,

(1) The philosophy is clear. An affordable speaker with decent amplification (and source) makes more sense than an expensive speaker with a cheap amp (and source).

(2) Amplifier power ratings: Watts per channel specs alone are useless . It is high current delivery (amps) that needs to be examined. Although it’s less of an issue now compared with the Jurassic era days (70’s) of valve amplification, if your speakers are demanding to drive, you’ll need a suitably muscular / beefy amplifier to support them.

Don’t look only at the headline power figure - see what happens when the impedance drops to four ohms. If the number nearly doubles, then your amplifier has good current delivery and will be capable of driving more demanding speakers.

https://youtu.be/th8CxTk22pY

https://youtu.be/Rtx2pYOS-dg

@akg_ca has it right. 

@jumia ….

Why do more powerful amplifiers impact speaker sound quality in a favorable way?

They don’t. Not always anyway. Several things should be considered. In example:

1. Amplifier power rating vs. speaker sensitivity

2. How does an amplifier handle impedance dips that some speakers present as a challenge to the amplifier. Some speakers with nominal 4ohm impedance may dip below 2ohms and not all amplifiers can or will handle it gracefully.

3. The size of a listening room and listening levels preference

Quality vs. quantity of watts and a proper matching based on the speakers you’re driving and size of your room are the most important considerations.
Just like with anything in this hobby, synergy is key.