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.  

Sounds loud. Our sense of loudness is largely based on distortion. The ear itself distorts at high levels so we naturally confuse distortion of the system with real loudness. Clean undistorted SPL can be very loud indeed and make the music breathe with effortless exhilarating dynamics just like a live performance - long before SPL becomes objectional for being too loud..

When you hear of a speaker being "discerning" of amplifiers, it often means that it is difficult to drive , usually in the bass region. 

These speakers prefer low output impedance amps. Rated power at 8 Ohms won't necessarily tell you this. 

Best,

E

+1 shadorne 

One of the best comments I got on my HiFi years ago was,

That's somewhat loud, but it really don't seem like it.

Driving a pair of Focal 1007Bes with a Rega Maia rated at 80W the speakers emmited a loud clicking rattle when I cranked them up too far, with the same amp into Thiel CS1.6s a sustained forte passage in Rattle and the Berlin Phil made the sound seem to sparkle and jump out at me from the sound stage, presumable when the power supply ran out of current (the Thiels were 3Ohn at areound 5KHz), A Naim 250, also 80W had no trouble with the same passage at even higher levels on the Thiels but it had a much stiffer power supply. The quoted power isn't the whole story.

+1 shadorne got it right with a minimum of words.
In addition, if a solid state amp is used, your tweeter may be at risk. A classic symptom of an underpowered system employing a solid state amplifier is fried tweeters.

It sounds like s... !

It took a lot of power to drive the bottom end of my known-to-be-hard-to-drive B&W N801's.  That was the biggest difference I can say was for-sure related to power output as I experimented with a Crown XLS-2502 which has tons of power.  It had other issues.  Switching ("permanently") to my Levinson 333 made all of the difference.

Just to agree with the above, it is the underpowered amplifier that blows the speakers. The clipping/distortion makes the speaker move in ways that it was not designed. Uninformed people think that it is too much power that blows the speaker. It is really the underpowered amplifier that clips and blows things us.
Remember that there are muscles in your ear that tighten up when exposed to loud volumes and decrease the movement of your eardrums and protect the delicate hair cells that would otherwise be damaged by high volumes.. That is why when you walk into a room the volume seems louder than it does a few moments later.

+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".