The only benefit to HDMI is that you don't need a second cable for audio. HDMI essentially combines a DVI connection with audio. Actually, because of this, you can acutally get converters that go from HDMI to DVI for a few bucks.
Because it already has a DVI connection in it, the quality over the HDMI connection would be identical. The only reason you'd want to use HDMI is if you'd like to run sound through the monitor--but it's not going to improve the video quality.
That being said, there's a ton of other factors that affect "video quality". Variations in display coatings, backlights, and LCD panels can make the same "red" icon look very, very different on two different displays. However, since DVI and HDMI use the same digital link, there's effectively no signal loss, so one wouldn't look any different than the other on the same display.