A TV is often used for less-interactive activities like watching movies, where buffering a few frames ahead so that it can interpolate between frames and make it look smoother isn't a big deal. However, when playing video games, this is undesirable input lag. I think most 120hz tvs have a "game mode" which avoids most of the input lag by skipping the interpolation and showing each frame a second time, making it little different from a 60hz tv in this mode. Also none of the standard TV connections (hdmi, vga, component, composite) are required to carry a full-resolution (1920x1080) at 120hz progressive. I've seen plasma tvs listed as being 600hz, but that's only internally. The video signal is still 60hz at most.
A 120hz monitor is usually designed to support 60 hz per eye with shutter glasses, but to support the 120hz signal at full resolution, it requires dual-link dvi or displayport (thunderbolt would probably also have enough bandwidth for 1920x1080 at 120hz, but so far there are very few monitors that have thunderbolt inputs at all). Most 120hz monitors also support using 120hz in 2D mode (i.e. non-stereoscopic) for smoother graphics. I have my doubts about the magnitude of improvement when using 120hz instead of 60hz, but I haven't actually tried a 120hz monitor (not counting crt monitors, which could only get that high a refresh rate at low resolution), so I could be wrong. If you can get to a physical computer store and compare them in person, that might give you a better idea.