120Hz HDTV and 120Hz PC monitors are not the same.
120Hz HDTVs improves video / movie image quality by insert interpolated frames between frames it receives from the video source whether it is Blu-Ray / DVD player or a PC's video card.
Basically what happens is the HDTV gets frames 1 and 2 from whatever source. The internal electronics compares the two, creates an interpolated frame based on the difference of the two frames, then inserts this "new frame" in between the original ones. So a Blu-Ray player feeds the HDTV frames 1 and 2, the HDTV electronics compare the difference between them and creates frame 1.5. It then inserts this interpolated frame between frames 1 and 2 to improve image quality and smoothness.
Next comes frame 3, the HDTV then compares frames 2 and 3, creates frame 2.5 and inserts. Next comes frame 4 and yada, yada, yada...
While this can improve image quality of video and movies, this process will create input lag for PC games. You need to look for a 120Hz HDTV that allows you to turn off this feature so that the HDTV will not create artificial input lag. I believe Samsumg HDTVs allows you to choose "gaming mode" to turn off this feature.
I have not done any research on 240Hz, but I believe it will add more post processing techniques to improve image quality and smooth playback. Thus, this will probably increase input even more because the electronics need to perform more post processing before the image is displayed on screen.