For you as an end consumer nothing if bandwidth is enough with either cabke, but techies like us will be sticklers that one is more open than the other, not bound by royalties and such.
That, and some video cards support higher resolutions over DisplayPort. For example, while my 2013 Mac Pro has both HDMI and Thuderbolt/Mini DisplayPort outputs, the only way to get 2160p60 HDMI output (without chroma subsampling) is via a DisplayPort-to-HDMI 2.x adapter (and I had to try about half a dozen different adapters before I found one that does this correctly).
Displayport usually has something like 50 % higher bandwidth compared to the HDMI standard of the same vintage. Which makes sense, since home video folks are fine with like 15 pictures per second and 4:0:0 chroma subsampling, but that doesn't really work for computers...
