They are fundamentally 2 different technologies.
A plasma works by trapping plasma (a 4th state of matter, but in this case considered a gas) in cells. When a current is applied, it lights up.
An LCD is essentially a filter that becomes transparent or opaque depending if current is applied to that pixel. Then there is a backlight behind it that supplies the lighting.
Plasmas have historically had nicer pictures, and were less expensive, but were more subject to annoying glare.
LCDs historically are brighter, less subject to glare, and more expensive, but not as pleasing of a picture.
However, I defy anyone to look at a Sony XBR Bravia LCD and tell me it's not the best picture in the store, plasma or LCD. This is the current TV I'm lusting after. Unfortunately, it's pretty much the most expensive TV out there for it's size right now.