On a CRT monitor, the refresh rate relates to how often the whole screen is refreshed by a cathode ray gun. This is fired down the screen at a certain speed which is determined by the vertical frequency set in your graphics card. If the refresh rate is too low, this can result in flickering of the screen and is often reported to lead to head aches and eye strain. On a CRT, a refresh rate of 72Hz is deemed to be "flicker free", but generally, the higher the refresh rate the better.
TFT screens do not refresh in the same way as a CRT screen does, where the image is redrawn at a certain rate. A TFT monitor will only support refresh rates coming from your graphics card between 60Hz and 75Hz (ignoring modern 120Hz monitors for a moment). Anything outside this will result in a "signal out of range" message or similar. The recommended refresh rate for a TFT is 60hz, a value which would be difficult to use on a CRT. The maximum refresh rate of a TFT is 75hz, but sometimes if you are using a DVI connection the refresh is capped at 60hz anyway and it will not allow you to select a higher refresh rate from your graphics card.
As a TFT is a static image, and each pixel refreshes independently, setting the TFT at 60hz does not cause the same problems as it would on a CRT. There is no cathode ray gun redrawing the image as a whole on a TFT. You will not get flicker, which is the main reason for having a high refresh rate on a CRT in the first place. The reason that 60Hz is recommended by all the manufacturers is that it is related to the vertical frequency that TFT panels run at. Some more detailed data sheets for the panels themselves clearly show that the operating vertical frequency is between about 56 and 64Hz, and that the panels 'typically' run at 60Hz (see the LG.Philips LM230W02 datasheet for instance - page 11). If you decide to run your refresh rate from your graphics card above the recommended 60Hz it will work fine, but the interface chip on the monitor will be in charge of scaling the frequency down to 60Hz anyway. The reason that some DVI connections are capped at 60Hz in Windows is that some DVI interface chips cannot scale the frequency properly and so the option to run above 60Hz is disabled. You may find that the screen looks better at 60Hz as you are avoiding the need for the interface chip to scale the resolution. Try it on both and see which you prefer, the monitor can handle either in many cases.
One thing which some people are concerned about is the frames per second (fps) which their games can display. This is related to the refresh rate of your screen and graphics card. There is an option for your graphics card to enable a feature called Vsync which synchronizes the frame rate of your graphics card with the operating frequency of your graphics card (i.e. the refresh rate). Without vsync on, the graphics card is not limited in it's frame rate output and so will just output as many frames as it can. This can often result in graphical anomalies including 'tearing' of the image where the screen and graphics card are out of sync and the picture appears mixed as the monitor tries to keep up with the demanding frame rate from the card. To avoid this annoying symptom, vsync needs to be enabled.
With vsync on, the frame rate that your graphics card is determined by the refresh rate you have set in Windows. Capping the refresh rate at 60hz in your display settings limits your graphics card to only output 60fps. If you set the refresh at 75hz then the card is outputting 75fps. What is actually displayed on the monitor might be a different matter though. You can measure the internal frame rate of your system using programs like 'fraps' and also some games report your frame rate. Remember, the frequency of the monitor is still being scaled down to 60Hz by the interface chip. If you are worried about frame rate in fast games then it is a good idea to try the refresh rate at 75Hz and see if you think it looks better. A lot of it could be based on placebo effect though, and if you have a decent graphics card which can handle a constant 60fps it might look just as good as if it were outputting 75fps. See which one you prefer. In some cases forcing a higher refresh rate (even above 75Hz) is possible, but this can have mixed results. In many cases frames are dropped anyway and so it makes no difference to the end result.
One other thing to note for Overdrive (RTC) enabled monitors is that running a TFT outside of it's recommended refresh rate can sometimes lead to a deterioration in the performance of this technology and the panel responsiveness is adversely effected! Read the details here.
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