120Hz HDTVs creates and inserts 1 interpolated frame in between two actual frames from the video source to "improve image quality" and smoothness. Example: Blu-Ray player send frames 1 and 2 to the HDTV which then creates frame 1.5 and inserts it in between frames 1 and 2, and lastly the frames are displayed on the screen. Next the HDTV receives frame 3, the HDTV then creates frame 2.5 and inserts/displays the frames on screen.
240Hz HDTVs creates 3 frames for every two actual frames it receives. Therefore, in between frames 1 and 2 which the HDTV receives from the Blu-Ray player, the HDTV will create frames 1.25, 1.5 and 1.75. This "improves image quality" and further increases smoothness.
The more frames the HDTV must create, the higher the input lag because the HDTV requires more time to create the frames.