Sharpness is a property of the pixel density, which is separate to the number of pixels (the 4K part), so it's not really a question which makes sense.
For example I have a 4K screen for coding but it isn't particularly high DPI because it's physically large at 68 cm so nothing on it is 'scaled' beyond normal.
And of course, for the same monitor size, compared to 1080p, it would be ~2x the density, so that's a valid fixed comparison to make too.
In other words, read the parent's question as: "if it worth it to go 4K over 1080p/2K for the same monitor size? Is it noticeably sharper?".
To anyone considering one, make sure it supports 60hz at full resolution, otherwise the choppyness makes it unusable.
One was a 15" laptop screen. It gave me no end of trouble. Anything that didn't scale properly would be unusably small due to the downright excessive pixel density, and a distressing number of apps didn't scale properly. Most everything else about that computer was great, but I ended up replacing it within a few months, mostly due to dissatisfaction with the display.
The other is a large external monitor. I can't remember how big - maybe 27"? I like it quite a bit. Combined with a tiling window manager, it's a bit like having a multi-monitor setup, only with even more flexibility in how to arrange your workspace.
So, I guess, my advice would be to ignore the marketing terminology and just make sure that your monitor has an appropriate pixel density - not too little, not too much.
It's definitely been worth it for me... I wish my work would let me bring my own... I'd rather the one 40" than the two 22" (1080p) that I have now.
After getting used to the retina display of macs, you realise how bad, how pixelated 1080p looks on texts. Especially the workplace monitors since they are usually the cheapest ones. Upgrading 4K gives you joy after being exposed to those cheap screens. :)
I use a LG 27UD69P-W 27" 4K at home and 27"@4K is the sweetspot for me.
The dpi is high enough that fonts are crystal sharp without any noticeable hinting artifacts as I've gotten older poorer screens give me a literal headache.
I don't have any scaling issues, the added screen real estate is a life saver.