There are two big differences:HDMI incorporates content protection called High Definition Content Protection (HDCP). HDMI supports audio in addition to digital video. (DVI only supports digital video)
Edit* On picture wise, they are identical. You might call HDMI as ajust newer version of the DVI.
Today I nearly bought a load of stuff to bring my 5-year-old Windows box up to date, but then I remembered that games are all shit these days and the people in multiplayer are all sociopathic wankers, so I didn't.