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With DVI output, you are sending the digital video signal directly from the video card to the monitor. This results in much better image quality because there is no in-between image processing. VGA, on the other hand, is an analog signal that has to be converted and therefore some quality is lost. VGA is also much more susceptible to distortion from outside influences, like power cables.
Where you can really see the difference between DVI and VGA is at higher resolutions. In a lower resolution, such as 1024 x 768, you might not notice much of a difference at all. This would be just fine on a 15” or 17” monitor. However, if you were to hook up a 22” widescreen monitor with a resolution of 1680 x 1050, the difference is quite noticeable. On a VGA connection, that 22” monitor is going to look kind of fuzzy and text won’t be so sharp. On a DVI connection, everything appears much cleaner and is easier on the eyes.