It all depends on what you are using the monitor for...

LCD (liquid crystal display) monitors are good for saving space on your desk and ok for running programs like Office and older games as the response time (normally Milliseconds on an LCD) is not as fast as a CRT monitor(with its reponse time measured in nanoseconds), however they tend to suffer from image burn slightly less than a CRT monitor. LCD monitors also use less power than a CRT eqivilent

CRT (cathode-ray tube) monitors are the best to use for graphic design software as they are the only monitors which display pure black, they are also better for fast paced games or applications which use a high refresh rate 95Hz +. CRT monitors suffer from image burn slightly worse than an LCD monitor in that the image has to be still for less time to create image burn than an LCD monitor. Radiation will be more form this monitor. But not so much you will be glowing green in a week. There's alot more radiation in your mobile phone / microwave etc than will ever come out of a monitor

Unless you are a serious gamer on the edge of technology or a serious graphic designer/artist, you wont really notice much difference. An LCD display will be brighter than a CRT but the picture may be a bit laggy under some conditions because of the way the picture is displayed, also the max resolution will be lower with a lower refresh rate. CRT's are cheaper with better display charactistics, but use more power, room and are heavy, but will have a higher resoultion and refresh rate.

Oh and the technology used in LCD's is getting better all the time so in the near future LCD may be the ONLY way to go .... I personally however would stick with my CRT anyday..