Display port and DVI are roughly the same but displayport carries sound, HDMI is more used for T.V. which carries video and audio.
From politicalslug post from Nvidia's forum:
HDMI at its most advanced spec, 1.4a, can output 1080p at 24 fps to each in 3D or 60 fps to each eye at 720p. Dual Link DVI is found on all modern video cards and 3D vision certified cards supports 1080 at a full 120HZ for 60 fps to each eye in 3D. HDMI is a lower bandwidth solution designed for televisions. HDMI also has television specific features like a remote control carrier signal. Displayport on the other hand is a ground up digital display technology that can also carry data and audio and is in every way a PC user cares about vastly superior to either DVI or HDMI.
Basically you want to use DVI or Displayport for a monitor and HDMI for a TV, and NOT use HDMI for a monitor.
Quality-wise they are all digital and give the same quality. As fantastik pointed out, max fps, data and audio are different features of each. If you only have 1 monitor without 3D or audio, any will get you 1080p60 exactly the same.