You could connect both but it shouldn't make any difference since both DP and HDMI are digital video formats. The same pixels are being sent to the monitor either way.
TVs will often treat HDMI differently. They assume HDMI is TV video, and will apply all sorts of processing to beautify a video image (which you don't want when displaying text and windows). They do not do this processing with DP/DVI/VGA. If your monitor has these video processing features, then using the HDMI connector might make a difference. But in all likelihood the picture will be the same with either DP or HDMI.
The program you use to play video on your computer may have image processing options. Streamed video generally doesn't. But if you're playing video files that you've downloaded onto the computer, then I suggest using VLC as your player. It can use ffdshow as its video decode engine, which has a plethora of image processing options.
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