Why do speakers improve with more powerful amps?

So, if I have a solid state amp that more than adequately powers a speaker, why do people recommend a larger more powerful amplifier to improve things?

Why do more powerful amplifiers impact speaker sound quality in a favorable way? Is it because more power is reaching the speakers? Mid and Tweeter drivers I was told receive a reduced signal versus bass drivers which receive relatively more power via crossovers.  All for the purpose of balancing a signal going to the various drivers.