The biggest advantage of prime lenses is their speed. The quickest zooms will go is f/2.8, but you can get normal and short tele primes at f/2 and f/1.4 which is a ton faster, and lets you shoot with natural light where an f/2.8 would need a flash. They're also ultra-small, ultra-light, and usually pretty affordable except for the more exotic ones. The fixed focal lengths force you to explore different angles and perspectives that you might not have found if you were using a zoom. I think zooms make me lazy, and I like the photos I get that I take with primes more, just because I find the angles and perspectives to be more interesting. The large aperture can also give you a very tight depth of field which is ideal for portrait type photos where you're trying to blur out the background and isolate the subject.
Two quick examples...
Father of the Bride, Nikkor 85mm f/1.8D @
f/1.8 on my D80. That's in a kitchen with an ugly refrigerator and stupid post it notes behind him, but you can hardly tell thanks to the tight depth of field, lol. With a zoom which wouldn't give as tight of a depth of field, more of the background might have been in focus which could have ruined the shot.
My 9 month old teaching herself to stand and balance with my Nikkor 35mm f/2 @
f/2 and 1/30s via natural light and Auto ISO at 560. To pull this off with an f/2.8 zoom, I would have needed iso1120 which would have been a lot noisier. Or with a slow f/3.5-5.6 or f/4 zoom that would have been at about f/4 wide-open, I would have needed over iso2000 which just wouldn't happen short of having a significantly more expensive body. No processing on this one - straight off the camera.
Another from my Nikkor 35mm f/2. This one is at 1/15s, f/2.8, and iso1000. That's my hand in the back, my wife's hand, and then our little girl's. This illustrates the great handling of prime lenses, since I had my D80 with the 35 f2 in the other hand. A much bigger and far clumsier f/2.8 zoom would have made this shot impossible, on top of the fact that they usually don't focus this close. Primes let you get right into your subject without a big honkin' zoom getting in your way.
So yeah, I'm a big prime fan.

As far as sharpness goes, the best professional grade zooms out there will match or beat primes for sharpness these days, but "in most cases" when comparing consumer grade zooms to similarly priced prime lenses, the prime lenses will definitely have better image quality more often than not.
One last one with my 50mm f/1.8 @
f/6.3 in Program Auto mode. This is the sharpest and contrastiest lens I have, and it only cost 100 bucks! LOL Can't beat it!