A little deeper on amp power please....

If somebody could elaborate on exactly how a higher watt amp will improve the sound of speakers (lower sensitivity speakers that “need” power).  More specifically, I get that when the nature of the recording and the volume setting demand an immediate spike in power, an amp that delivers the spike will perform better than one that does not.  But when I used to have an amp with output meters, it would be in single digits for most normal listening, and I don’t recall what a spike would have been - I want to say 15 or 20 watts.  What I am scratching at is whether there is something more to power, i.e. the notion that the effortless power of, say, a 300 watt amp would somehow be an improvement over an otherwise similar 75 watt amp…even if a spike is just 20 watts.  Hope the question make sense.