Not enough power what does it sound like?

+1 @shadorne

Muisc always sounds louder with an underpowered amp. Folks are sometimes tricked into thinking their underpowered SET tube amp has enough juice to drive their 88-92 dB speakers because it plays them loud. Well it may be a good match or it may well be the added noise from distortion.

Not enough power = clipping.  

https://en.m.wikipedia.org/wiki/Clipping_(audio)

Avoid it at all costs. 

Excellent analysis above; I have painful, expensive, agonizing experience over-driving amps that resulted in damaged tweeter voice coils

though my system doesn’t sound ‘loud’, I know it is as soon as I try to engage in normal conversation; it’s clean, it’s dynamic, it’s a joy to listen to, but I really don’t sense how loud the system is once I try to talk with some one

I could not bear the loudness if it didn’t produce a clean reproduction and that requires sufficient power, more is better - and safer 

+3 for @shadorne. I’ve never heard lows sound that bad, but highs and midrange will sound grating to you. As was said, they will sound loud and fatiguing. As @hifiman said, one can never have to much power,  and current, not necessarily watts, is what you’re after, of course they can come hand in hand. I think it takes 750 high current watts to reproduce the canon shot in the 1812 overture. Brief transients of an amp trying to reproduce what you may be asking it to, then because they can’t and they start clipping the top of the sine wave, can and will fry your protective crossovers and tweeters. A good driver can take much more power than it’s rating, if the power is clean.  Those snare cracks and cymbal crashes will drain your amp dry. 

Whenever I've upgraded to a more powerful amp, it's also been an upgrade to a higher priced unit with more sophisticated design and construction, so my observations might not be applicable, if you are comparing two amps of similar quality but differ only in their power output.

Having said that, for each upgrade I've found the system sounds like it has more "air" around it, the stereo image and sound stage is improved, and usually the bass is more defined.  I upgraded from a 35 watt NAD to a 180 watt NAD, and the new amp just didn't sound like it was exercised that much, sort of like the lower powered unit was "struggling".