I don't believe damping factor has been mentioned. Higher power amps tend to have a higher damping factor. With bass in particular, a higher damping factor provides better control of the driver and depending on the impedance of the speaker can result in more accurate reproduction.
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.