I'm not sure of exactly how each works, but plasma tvs have much higher contrast ratios. They generally have richer colors, and are better for gaming.
Some high end LCD tvs, like Sony Bravia, have rich color and high enough contrast to match some of the high end plasma tvs.
Plasma tvs have to be kept upright at all times. Laying them down, or carrying the tv upside down could ruin the tv. Plasma tvs are shipped from the manufacturer with tilt gauges. If one has gone past the acceptable tilt angel, the gauge will permanently show the evidence.
You also have to consider burn in when buying a plasma. This can happen when you pause your plasma tv on a movie or television show for too long, or from any television broadcast or video game that leaves game scores, tickers, network logos, etc... on your screen for too long.
LCDs are much easier to care for. They are generally lighter, and you don't have to worry about tilting them the wrong way. LCD screens have the advantage of not reflecting light, which makes for a seemingly clearer picture next to a light reflecting plasma screen.
Most plasma tvs now have some sort of non-reflective screen, and most also have some sort of pixel rotation or other method in place to prevent burn in.
If you are going to get an LCD, I would recommend the Sony Bravia lineup. If you'd rather get a plasma tv, I would recommend Panasonic or Pioneer.
I have a Panasonic Viera. I bought it before they added the non-reflective screen. I've been very happy with it. It looks better than much more expensive LED and LCD tvs that I've seen.
I've fallen asleep with videos paused before. I thought it would cause burn in, but it always goes away after I watch something else for a few minutes. I've done a lot of stuff like that and never caused any permanent damage.