DVI and HDMI can both handle high HDTV-like bandwidths. In Mr D's case, using a DVI cable might offer a slightly better picture, and depending on the monitor, it might send more details about itself to the PC.
Re adapters, the signals that are common between DVI and HDMI are electrically compatible, so adapters are cheap and plentiful. But you have to get audio from one device to the other using a different route, since there's no audio on DVI.