When you hook up your computer to a 1080p display (not for video, just using the computer), it looks pretty 2002. 4k fixes that. Heck, for still images at close range, I can see the benefit of >4k (
obligatory xkcd).
When you do non-native image resolution scaling, 1080p doesn't do terrible, but 4k does better due to the smaller pixels. This covers everything from your out-of-the-box overscan setting, to 21:9 displays to PC connections.
So on the display end, I see the benefit of 4k. I still have a 1080p display, but I've always been pretty sad about 1080p when I hook up the computer.
As for 4k content, comparing the same content, the difference is minor. But the content's not always the same. I have a 4k player and the UHD copy of the Big Lebowski. Does that film need 4k resolution and HDR? No way, but the regular 1080p Blu-ray is pretty underwhelming, and the 4k disk, scaled down to 1080p/SDR looks like the 1080p Blu-ray should have. I'm not sold on 4k media per se, but there are some cases where the comparison is apples to oranges, and the 4k version is clearly better for unrelated reasons (similar to SACD's sounding better than CD's not because the medium was superior, but because a better audio master was used to create them). Also, maybe some day I'll have a giant honking outdoor theatre projection system, and the differences that look minor now will become less minor. But I kinda doubt it.
But I'm not buying another display until they do a better job with motion handling. If we could get motion handling that could rival CRT or Plasma, that's the point where I'd replace a perfectly functional display. Otherwise I'm likely waiting until it dies. 4k displays are nice, but not enough. 4k players on the other hand give you access to some nicer disks.