If you have a 1920x1080 (1080p) 60Hz monitor the picture should be the same with either cable. It's when you get to higher resolutions, refresh rates, and one of the adaptive sync formats that there can be a difference.
For example, I have a 165Hz 2560x1440 (1440p) monitor with G-sync. It has an HDMI input but if I want to use the monitor's high refresh rate or G-sync features I have to use DisplayPort. HDMI doesn't support framerates that high or G-Sync. I'm not positive on this but I think that is changing with HDMI 2.0. For that to work, both the monitor and GPU would have to support HDMI 2.0 though.