No HDMI and DVI are both digital signals there will be no degradation in quality between the two. If in fact they are compatible with each other as referenced on 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.
Both are much better than VGA. VGA is an analog signal. There are noticeable differences when using a VGA connection vs a pure digital one. Everything will appear to have a slight fuzziness, it's not horrible but if you have good vision you can see the difference up close. With pure digital, the text, images, etc with all be sharper. This is a side effect of the analog to digital conversion that the LCD monitor has to do. Additionally as part of that conversion process most LCDs have an auto correction button to dynamically adjust the VGA signal to fix over-scan and other minor adjustments to the VGA source. Something that you don't ever have to do with a digital source. I would avoid buying a monitor that only has VGA input, spend $20 to $50 more and get a better display that has at least one digital input source, either DVI or HDMI.
Confirming Source for LCD Analog vs Digital inputs.
I have had bad luck with 2 Mini-DVI to VGA adapters purchased from Monoprice.com that had Red, Green, Blue hewing due to poor wiring. For example, I had to slightly twist the cable to get the picture color just right. They replaced one and then the second one did the same thing, I am due for a third try... However my mini-DVI to HDMI adapter from them works flawlessly and has since day one.