Not enough power what does it sound like?

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.