Video playback looks better with a higher scan rate. A lot of content that was designed for TV (EG almost all historical documentaries) is going to be 25Hz interlaced (UK and Australia) or 30Hz interlaced (US). If you view that on a low refresh rate progressive scan display (EG a modern display at 30Hz) then my observation is that it looks a bit strange. Things that move seem to jump a bit and it’s distracting.
Getting HDMI to work with 4K resolution at a refresh rate higher than 30Hz seems difficult.
What HDMI Can Do
According to the HDMI Wikipedia page , HDMI 1.3–1.4b (introduced in June 2006) supports 30Hz refresh at 4K resolution and if you use 4:2:0 Chroma Subsampling (see the Chroma Subsampling Wikipedia page  you can do 60Hz or 75Hz on HDMI 1.3–1.4b. Basically for colour 4:2:0 means half the horizontal and half the vertical resolution while giving the same resolution for monochrome. For video that apparently works well (4:2:0 is standard for Blue Ray) and for games it might be OK, but for text (my primary use of computers) it would suck.
So I need support for HDMI 2.0 (introduced in September 2013) on the video card and monitor to do 4K at 60Hz. Apparently none of the combinations of video card and HDMI cable I use for Linux support that.
The Wikipedia page alleges that you need either a “Premium High Speed HDMI Cable” or a “Ultra High Speed HDMI Cable” for 4K resolution at 60Hz refresh rate. My problems probably aren’t related to the cable as my testing has shown that a cheap “High Speed HDMI Cable” can work at 60Hz with 4K resolution with the right combination of video card, monitor, and drivers. A Windows 10 system I maintain has a Samsung 4K monitor and a NVidia GT630 video card running 4K resolution at 60Hz (according to Windows). The NVidia GT630 card is one that I tried on two Linux systems at 4K resolution and causes random system crashes on both, it seems like a nice card for Windows but not for Linux.
Apparently the HDMI devices test the cable quality and use whatever speed seems to work (the cable isn’t identified to the devices). The prices at a local store are $3.98 for “high speed”, $19.88 for “premium high speed”, and $39.78 for “ultra high speed”. It seems that trying a “high speed” cable first before buying an expensive cable would make sense, especially for short cables which are likely to be less susceptible to noise.
What DisplayPort Can Do
According to the DisplayPort Wikipedia page  versions 1.2–1.2a (introduced in January 2010) support HBR2 which on a “Standard DisplayPort Cable” (which probably means almost all DisplayPort cables that are in use nowadays) allows 60Hz and 75Hz 4K resolution.
Comparing HDMI and DisplayPort
In summary to get 4K at 60Hz you need 2010 era DisplayPort or 2013 era HDMI. Apparently some video cards that I currently run for 4K (which were all bought new within the last 2 years) are somewhere between a 2010 and 2013 level of technology.
Also my testing (and reading review sites) shows that it’s common for video cards sold in the last 5 years or so to not support HDMI resolutions above FullHD, that means they would be HDMI version 1.1 at the greatest. HDMI 1.2 was introduced in August 2005 and supports 1440p at 30Hz. PCIe was introduced in 2003 so there really shouldn’t be many PCIe video cards that don’t support HDMI 1.2. I have about 8 different PCIe video cards in my spare parts pile that don’t support HDMI resolutions higher than FullHD so it seems that such a limitation is common.
The End Result
For my own workstation I plugged a DisplayPort cable between the monitor and video card and a Linux window appeared (from KDE I think) offering me some choices about what to do, I chose to switch to the “new monitor” on DisplayPort and that defaulted to 60Hz. After that change TV shows on NetFlix and Amazon Prime both look better. So it’s a good result.
As an aside DisplayPort cables are easier to scrounge as the HDMI cables get taken by non-computer people for use with their TV.