HDMI support 10bit per channel, but it's unlikely that even if your monitor supported it that you would be able to get it to work, let alone notice the difference. As mentioned before for all intent & purposes they are the same for straight video.
HDMI-> Cleaner setup with Audio/Video in one cable. HDMI 1.3 has higher bandwidth than DVI
All my PC video interface use HDMI.
If the DVI cable is already available with your monitor then use it. Don't need to spend $ for an HDMI.
If you are to spend money for a video cable---> HDMI is the preferred interface