Consistently good colors. Sometimes even PC monitors won't negotiate properly with the PC and start displaying washed out colors. IIRC this had to be forced on a Raspberry Pi on the PI side. Not all monitors can be adjusted.
Consistently no under/overscan. For some reason, the TVs we had in conference rooms figured it would be smart to cut the borders of the image and zoom in, so you get missing bits and whatever's left is a blurry mess.
For the TV situation, you usually don't have a full remote to adjust it, if it even supports that. You often don't have time to look up things in the TV's typically crappy menu system. I've usually found the option to disable over scan, but using full-range RGB seems less common.
Also, HDMI seems to lag DisplayPort capabilities when it comes to higher resolutions and refresh rates. When my 2013 MBP came out, it could drive a 4k@60 screen over DP. HDMI required the 2.0 version to do that, which, IIRC, came much later.
I think that one very much depends on when you chose to look at it. HDMI 2.1 has more bandwidth than DisplayPort 1.4, enough to do 4k@120 which DP1.4 wouldn't be able to do it without dropping color down to 4:2:0. DisplayPort 2.0 devices are starting to come out, but even Nvidia's RTX 4000 series still do not have DisplayPort 2.0 (but do have HDMI 2.1). While TV started supporting HDMI 2.1 around 2019, with the PS5 and Xbox Series X having HDMI 2.1 ports.
So while DisplayPort may be ahead now with DisplayPort 2.0, HDMI was ahead for at least 4 years with HDMI 2.1
Although, if I'm not mistaken, my particular PC monitor initially didn't support HDMI 2.0, even though it's a 4k panel. There was a further revision which included it. I have that revision, but support is still somewhat wonky, in that it can't seem to switch on its own from 2.0 to 1.4b.
I've only encountered overscan on TVs not monitors, but I don't give conference room presentations, which would be very annoying. On Macs there compensation for that.
I've had trouble with colors where it uses YPbPr rather than RGB, but that seems to be an Apple thing where it's done on purpose for non-Apple-approved displays and it happens for both HDMI and DP. Generating a custom EDID profile fixes that. Can't recall having trouble with color range, sometimes the display has a setting but the default always looked better to me.
I've used 4k@60 HDMI just fine (and 4k@30 on an early Apple adapter), but more often use 1080p anyway. I use USB-C with my 4k displays which likely runs DP on them.
Right. So... DP is better than HDMI?
The color issue I had was not with a Mac but with an HP laptop on an HP monitor. My understanding is that there's something about "broadcast colors" or something, which is "regular" RGB only with a narrower range. I think the PC thought the monitor was a TV with a limited range, which the monitor was not.
With the TVs I plug into, it's usually harder to judge since their color rendition tends to be all over the place anyway and tend to have the reverse issue (too much contrast).
I remember the overscan control on the mac, but it still was a PITA to have to fiddle with that instead of, you know, just plugging the screen in and being in business.
While I've also had numerous positive experiences with HDMI where things seemingly "just worked", I've never had an issue with DP. It always worked. Hell, even my gaming GPU, which came out a while after HDMI 2, and supported it out of the box, connected to my monitor with full HDMI 2 support, still has weird colors compared to DP. No tweaking in the AMD drivers managed to get me the proper output, so I went and bought a cheap Chinese DP KVM instead. Which worked with no fuss.
All this makes me automatically pick DP if given the choice, and discount any computer or screen that only does HDMI. Which makes it pretty tough to buy a TV, so I just watch movies on my computer monitor.