I think the main thing for me, going from a TN panel computer monitor, to a 32" HDTV is definitely the input lag. There's probably better displays than the one I'm using, but I think that would be the main issue with most TV's you would buy today. Color and clarity are good, dot pitch won't compare to some of the pricier large computer monitors but it's more than good enough to live with.
It's just when you play those twitch shooting games and such, the input lag on my TV isn't bad for anything else (I have used both PC and Game mode with similar results).
I wouldn't worry about 120hz processing for a TV used as a monitor unless you're watching movies. If it's easy enough, you can just change your settings off game modes to watch videos from your computer. I wouldn't want 120hz processing for any games, fast moving stuff would probably produce artifacts.
But definitely, if I were buying a display now (I got my current one as a gift), input lag would be my number one selling point. I'm pretty sure the Samsung B650 or A650 series are decent when used in Game mode.