HDMI and DisplayPort are used for different purposes. They are sometimes used for gaming but between the two, DisplayPort is considered to be better. It will provide you with games that will not lag as much as you have expected. Take note that HDMI and DisplayPort are not compatible with each other so you cannot use them together. You can just choose one over the other. HDMI will offer an Audio Return Channel while DisplayPort will not offer this. When it comes to their image quality, you will almost not see any difference. There will just be changes that are available based on the available version.
There is one similarity between the HDMI and the Display Port. They are both high-speed digital interfaces. There are some things wherein they are highly different from each other. You can expect to see the HDMI in different consumer electronics that you will find right now. You can expect to see the Display Port in different computers and all of the other peripheral items that you need to use. For example, when you see some video monitors that you need to connect to the computer, you can expect that these items will make use of the Display Port as compared to the HDMI. Analog products will most likely use Display Ports as well.