Not enough power what does it sound like?

Music is uninvolving, two dimensional, homogenized, textureless, etc.

The sound will be distorted and dynamically depressed.  The sound will distort on louder transients such as a dynamic drum solo.  If the amplifier is driven into clipping on loud transients, you may even hear a "snapping" sound.  The snapping is the speaker reproducing  an improperly formed sine wave, The snapping or consistent clipping can and likely will damage your speakers!!

Audio Axiom:  You can never have TOO MUCH power!

Assuming you're referring to a system with a SS amplifier:  Slowly increase your listening volume and focus your attention on the middle(especially vocals) and higher frequencies.  They will start to sound distorted and very irritating(clipped off/square waves/odd order harmonics).  Don't leave the volume there, as most tweeters can't take much of that abuse without voicecoil damage.  Crossovers see the high freq distortion as DC and pass it right to the tweeters.   Tube amps clip much more gently(rounding off the waveform, rather than chopping it off) and are somewhat harder to gauge, when being overdriven.

I think the real question is what if you don't have an amplifier capable of producing enough current.  
Running out of power, you'll find that your amplifier is going into clipping and producing higher distortion levels when it gets at its limits,  if this happens before you get loud enough, you don't have enough power. 

Current on the other hand in what you hear is dynamics, flow etc.  
I think the reason that we relate current with power is that more powerful amplifiers typically have a better or heftier built power supply that can deliver the current needed.  
I recall years back at Marcof Electronics.  We were testing a 100 watt power amp,  It was ok, played loud enough, but just didn't have any life.  We had a mock up of a 40 watt amp that had 3 times the power supply that the 100 watt amp had and bam,  the differences were not subtle.  
Today,  I think of a Clayton S40 or a Coda S5 amplifier.  These amps have the current to drive just about anything and if you have speakers that stay within their volume capabilities,  these amps are examples of high current lower powered amplifiers. 
I Hope this helps.