They're both digital so the image quality is the same for same settings. Both types have their ups and downs. DP supports higher refresh rates than HDMI 1.x, HDMI 2.0 on newer displays matches DP on that. There's generally no audio on DP monitor connectors, something to be wary of. Buying DP cables can also be a minefield, all cables are not equally compliant, as there is an issue with an old power connector on many displays. You'll basically be forced to buy "certified" cable brands, which obviously adds ten bucks to the price tag. Those cables doesn't connect that troublesome power pin.
Many displays (all with HDMI 1.x?) require a DP connection for adaptive sync like FreeSync.