This question seems to always be in the minds of many that buy amplified audio gear. While not an expert by any means, I have had many different setups with different power amps and receivers etc. IMHO, amps beyond the first as was stated by Tazishire above, can make a difference in sound quality and levels to the extent that speakers are relatively inefficient in generating sound waves at the same level as those recorded in the master disc, tape, LP, etc. and this creates a demand on the speaker which will draw more power from the amp in an attempt to keep up with especially dynamic music passages. In the old days we called this headroom for the amp/speaker system. Headroom as I understand it, is the "reserve" amps available to the speakers during changes in recorded volume levels. Headroom is usually a fleeting demand on the amp but lower wattage amps have less headroom for the speakers to draw on to create the louder passages.
When I buy equipment my goal is to try to match amp watts with speaker design (two way, three way, highly efficient, less efficient,), intended use (rock vs chamber music, etc) and room size and treatments (big empty room, one with lots of sound dampening, small live room, etc.); all these together have an effect on the amp/speaker system and the amount of useable amps for headroom becomes an important factor.
That said, the quality of the source, amps, switching, wires, and speakers can all play a part in the quality of the sound reproduction. A lousy receiver, preamp or amp can all create noise in the signal and more amps only make it louder (within limits). My philosophy is to by the best sound reproduction system you can afford and one with sufficient amps to meet your needs. Low watts on a great sounding amp can be far better than 400 watts on a crappy one.