The advantage of using a telescope in space is that you don't have to look through the Earth's atmosphere. For very detailed observations the atmosphere is pretty murky and horrible so it's a real advantage to get above that. You've probably seen HST pictures, and they really are much more detailed than you can get from the ground.
The disadvantages are mainly to do with the hassle of operating in space. It's much more expensive, so you can't have such a large telescope. If things go wrong it's much harder to repair them. You can't update the instruments so often so they quickly become out of date. Also with the modern technique of Adaptive Optics (basically correcting for the turbulence of the atmosphere as you observe), ground based telescopes are catching up with the HST.
By the way, the above is for optical telescopes which I assume is what you mean. For other wavelenghts there is no choice as our atmosphere can block them completely (eg. Far infra-red and X-rays and Gamma-rays). Telescopes for these have to be in space. For most radio wavelengths the atmosphere is very little problem, so instruments like Arecibo and the VLA are not limited by the atmosphere at all.
This page was last updated July 18, 2015.