The advantages of higher frequencies depend a lot on what you are viewing. In most resolutions, as per ATSC standards, the signal your TV will receive is likely to be either 24 or 30 frames per seconds, as the 60 frames per seconds format is not yet standardised to all display resolutions (as per ATSC standards, 60fps is supported on 720p resolutions, but not yet on 1080p).
If your television has a 60hz frequency (which is the standard in North America to be in sync with the AC current provided), this means that a signal sent at 30fps will display each frame for exactly 2 60th of a second. This is very good, as each image will be displayed for the same duration. However, if the signal sent to your television is set at 24fps, this means that half of the images will be displayed 2 60th of a second while the other half will be displayed 3 60th of a second. This is what is called a 2:3 pulldown. In slow shots with a steady camera, this can result in telecine judder.
In fast moving shots, this judder might be le
ss visible, but it can reduce the overall sharpness of the image, leading to motion blur.
Most motion pictures are still shot using 35mm film at 24 frames per second, as this film type has a very good definition. In older televisions, this lead to a large use of the 2:3 pulldown. However, modern HDTVs also use what is known as motion interpolation to generate additional images in between the frames to artificially raise the footage to 60fps. On a 24fps footage, the 2:3 pulldown is first used to raise the signal to 30hz, then interpolation is used to raise it further to 60hz. This reduces the judder slightly, but it does little to alleviate other sources of motion blur, such as your TV's pixel response time, resolution resampling or compression algorithms used in some video formats.
To alleviate those other sources of motion blur, TV manufacturers have begun making TVs with higher refresh rates. As 120hz is a multiple of both 24 and 30, it can eliminate the need for the 2:3 pulldown. In effect, each frame of a 24fps footage would be displayed 5 times before switching to the next, eliminating judder. However, a lot of 120hz TVs are currently using their faster frame rates only to reduce motion blur on regular 60hz signals by using more motion interpolation. On such TVs, the 2:3 pulldown is used to convert the 24fps footage to 60hz, and only then does the motion interpolation take place. This double conversion of the original signal can lead to visual artefacts in the end signal, reducing the quality of the image in a way that is probably assimilated to motion blur although it does not stem from the same source.
This HDTV Magazine article gives good complementory information on the subject.
My personal conclusion would be that a good 120hz TV can in fact be better than a 60hz one. However, I see no need for a 240hz TV as it does nothing more than a good 120hz one to reduce the need for the 2:3 pull down. The only thing it would do is add more interpolation to the footage, which can result in unnaturally smooth motion and an equally unnatural cardboard cutout look (characters appear as cardboard cutouts in front of the background).