The higher the resolution the higher the details:
In theory, yes
In practice, the higher resolution costs so much power and space that often texture quality has to be reduced at the same time, thus, while having more details than HD for instance, 4k often doesn't look nearly as great as it's supposed to be.
Also, if your Screen/Monitor is too small, the increase in resolution will be barely discernible, while getting higher detail textures will be much more visible. Just look at the picture and you see just how much they had to zoom in to make the difference obvious.
The Nintendo eShop rating Thread: http://gamrconnect.vgchartz.com/thread.php?id=237454
The Steam/GOG key gifting thread: https://gamrconnect.vgchartz.com/thread/242024/the-steamgog-key-gifting-thread/1/