IMHO there is only a subtle and subjective difference between shooting 4K vs HD and outputting to HD.
I recently did one test with my P3P (video shot while she was on the ground) and had to look for the difference on still captures at 100-400% or video on 6MB monitors *.
* Shooting 4K, then output to HD produced smoother (and blurrier) borders compared to HD. On the other hand, shooting HD produced more contrast to the borders. I didn't really know which one I preferred, though.
Editing 4K is more resource-hungry.
Watching 4K is also resource-hungry. I compared 4K and HD output on a 6MP monitor, a speedy Windows 7 PC and VLC player. Yes, 4K was crispier with more detail. But even then I had to look for the difference and ignore the real video content... And even on that speedy PC, the 4K playback was slightly but disturbingly stuttery while HD playback was smooth. I haven't tested if there are better players than VLC.
That said, I still shoot 4K (3840 x 2160 4K UHD because it is 16:9 unlike 4096 x 2160 4K DCI which needs 107% zoom-in for HD output with no black borders) because it allows to straighten the horizon the usual ~1° with more headroom for the output (I haven't yet tested if this really counts, though).
If/when my P3P breaks, I will re-consider if 4K camera is worth the price then.