I recently upgraded to a 1080P monitor and thought I would try an HDMI cable instead of DVI. What I discovered is that when you hook up the HDMI cable, the monitor settings begin to work like an HDTV, rather than a PC monitor. Different TV resolutions became available in the control panel, but the native one didn't fit the screen.
The most annoying thing for me was, my monitor no longer went into Standby mode when I shut the PC off. I had to manually turn the monitor off and on every time, since it was working like a TV. Needless to say, I switched back to DVI so that my monitor resolutions are correct, and my monitor now goes properly into Standby mode when I shut off the PC.
In terms of video quality, there is no difference between HDMI and DVI. The only real benefit of HDMI is that is also carries a sound signal, if you require that for a home theater or if your monitor has built-in speakers.