The difference between VGA and DVI is pretty simple.
VGA is a digital signal (think 1's and 0's) that is converted into an analog signal (think wavy lines) to be passed to your monitor, which then converts that back into digital.
DVI is digital from start to finish.
As Mindover said, this matters when you start trying to use higher resolutions. Because the analog signal in VGA is susceptible to interference (e.g from the cable, from outside signals, etc), you get a less precise picture. DVI, being all digital, doesn't suffer from this problem.
On anything below 1920x1080 you'll struggle to see any quality change between VGA and DVI, but anything above that and you'll probably start to see the difference (depending on your individual circumstances of course).