Is there any benefit of using HDMI over DVI?
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.
HDMI shouldn't cause a difference in image quality vs DVI. However, you will have to worry about bandwidth issues. These issues really only occur if you're intending on playing computer games. Over HDMI, I found that some games I tried to play would cap the refresh rate at an abysmal 24 Hz, which translated into a lot of image tearing and headaches. However, I found that it was perfectly fine for typical desktop/office tasks and movie watching.
If that's all you're doing, then you're perfectly fine using HDMI. If you plan on gaming, though, I'd recommend keeping an eye out for performance issues. They should be readily apparent.
*Note that these issues that I described can also depend on the software you're running and the drivers you have. My experience may not directly translate to yours. Ultimately, the best way to find out is to try it yourself and see what happens.
HDMI vs. DVI
Here are a few other facts beside the audio being included with HDMI.
From Wikipedia
HDMI is backward compatible with single-link Digital Visual Interface digital video (DVI-D or DVI-I, but not DVI-A). No signal conversion is required when an adapter or asymmetric cable is used, so there is no loss of video quality.
This means that HDMI is not the exact same as DVI Dual link whose bandwidth "is limited only by the bandwidth limits of the copper the DVI cable is constructed of and by the DVI signal's source." Also, there are several specification types and versions to HDMI that limit it's speeds.
With those two combinational factors there's a possibility that a HDMI cable and/or port will not produce the same quality as the DVI and vice versa. For example, with my home PC I ran into an issue with my HDMI cable connected monitor only running at 60Hz vs the 120Hz that I got from the DVI-DL. This can cause issues with 3D enabled content.
Note: There ARE HDMI Cables that can handle 3D content, but not all.
Here are some other sources to look at and a summary of their findings:
HDMI vs DVI: High Speed HDMI can support up to 4k x 2k res while DVI's max is 2560 x 1600
ecoustics.com:
So, which is better, DVI or component? HDMI or component? The answer--unsatisfying, perhaps, but true--is that it depends. It depends upon your source and display devices
The Introduction of the Adapter
Whenever you introduce any type of adapter, you are limited by the specifications of the lowest operating medium. For example, a VGA to DVI adapter is limited by the specifications of VGA. In your case (HDMI to DVI), the limitations are dependent on the individual devices being used. If the receiving monitor is a single link DVI then there is probably no difference in quality. However, if the receiving monitor is a dual link DVI, there may exist a reduction in quality.