Generally speaking both should provide the same video quality at 1920 x 1200 resolution or less as long as the quality of the components used are similar. If the components used for the HDMI connection is bad, then DVI will look better and vice versa.
HDMI can carry audio signals which is a plus for many people. However, it cannot be used drive a monitor that can handle higher resolution because there is no such thing as a dual linked HDMI connection yet to the best of my knowledge. That means two HDMI connections wrapped into one.
To drive a 2560 x 1440 or 2560 x 1600 resolution monitor you must use a dual linked DVI connection or two DVI connections.