DisplayPort and HDMI both tick a lot of boxes - they can game at 4K, perform office duties and play multimedia or streaming services without issue - but they both also offer something extra the other one doesn't.
DisplayPort is the choice for gamers running high-end graphics cards and high-end gaming monitors. Support for G-Sync and Freesync are dependent on DisplayPort, and the latest video cards already offer HDR and 8K over DisplayPort.
If you have DisplayPort available on both your PC and Monitor, your best choice is DisplayPort. It provides the most 'bells-and-whistles' features and the best possible gaming experience.
Most DisplayPort sources also offer Dual-Mode output, which supports passive VGA and DVI adapters. This gives it an advantage over HDMI when working with older PC screens and accessories.
The ace up HDMI's sleeve is its ability to offer high resolutions over longer distances than DisplayPort usually can. Where regular copper* DisplayPort cables only go up to 3 or 5 metres, a good HDMI cable can take 4K@30 up to 10 metres and 1080p up to 20 metres or more.
Being so common, HDMI is generally cheaper and has maximum compatibility with modern home theatres and presentation hardware in offices and auditoriums. HDMI is a great universal choice and a very capable fall-back for when DisplayPort isn't available.
HDMI also boasts a wide range of after-market accessories like switchers, splitters, audio extractors and converters which can make it operate with even older tech like Composite and Component Video over RCA jacks. This versatility makes HDMI a powerful choice that won't steer you wrong in most cases.