crt monitors offer more colors (a good one, not a crappy one) and have better contrast, give videos and games more depth, the best performance for gaming, video editing and any motion applications.
However, they are subject to screen flicker, irregular screen shape, distortion, and fuzzyness with higher refresh rates, have about half the lifespan of an lcd, lose their brightness and sharpness over time and give off a lot of heat, as well as use almost 80-120 watts or power most of the time.
They are huge, heavy and cause eye strain, give off radiation, are subject to magnetic interference, and take along time to warm up.
lcd's shine when it comes to electricity usage, size, weight, heat dissipation, text clarity, perfect screen geometry, better tonal consistency (especially white), higher text contrast (crt's are limited by the electron gun's video bandwidth) and color brightness, but are not too great for fast moving things or serious photo editing, can have stuck pixels, washed out colors and are more fragile.
But they look cool and more attractive than a big hunk of a crt, and use digital connections (all new video cards are forcing the move to lcd by only putting digital ports).