DVI, HDMI and Displayport all carry a digital signal, so picture quality will be the same for each type of cable.
HDMI can carry audio and video, but tops out at around 1920x1200 @ 60Hz. Later specifications of HDMI can run higher (the latest can do 4K @ 60Hz) but most of the HDMI ports in the wild are the earlier specifications, hence the lower resolutions possible.
DVI tops out at 1920x1200 @ 60Hz for single link, but can run much higher with a dual link connection. DVI cannot carry audio (some GPUs use the spare pins to carry audio for DVI/HDMI adapters, I think, but most screens don't support audio in from DVI).
Displayport can carry high resolution video, and can carry audio. Displayport also does not require a clock signal, where DVI/HDMI/VGA do - this is why AMD cards from 5xxx/6xxx/7xxx had to have the third screen on a DP output, the cards only had 2 clock generators built in.
The advantages/disadvantages of each type of connection only really come into play when you're running a screen higher than 1920x1200. For anything under that resolution, the only difference between DP/HDMI/DVI is that HDMI carries audio. Otherwise, use whatever cable you want and be happy.
