Using memcpy() when source and destination overlap can lead to undefined behaviour - in those cases only memmove() can be used.
But what if I know for sure buffers don't overlap - is there a reason to use specifically memcpy() or specifically memmove()? Which should I use and why?
memcpy() doesn't have any special handling for overlapping buffers so it lacks some checks therefore it is faster than memmove().
Also on some architectures memcpy() can benefit from using CPU instructions for moving blocks of memory - something that memmove() cannot use.
If you're interested in which will perform better, you need to test it on the target platform. Nothing in the standard mandates how the functions are implemented and, while it may seem logical that a non-checking memcpy would be faster, this is by no means a certainty.
It's quite possible, though unlikely, that the person who wrote memmove for your particular compiler was a certified genius while the poor soul who got the job of writing memcpy was the village idiot :-)
Although, in reality, I find it hard to imagine the memmove could be faster than memcpy, I don't discount the possibility. Measure, don't guess.
Assuming a sane library implementor, memcpy will always be at least as fast as memmove. However, on most platforms the difference will be minimal, and on many platforms memcpy is just an alias for memmove to support legacy code that (incorrectly) calls memcpy on overlapping buffers.
Both memcpy and memmove should be written to take advantage of the fastest loads and stores available on the platform.
To answer your question: you should use the one that is semantically correct. If you can guarantee that the buffers do not overlap, you should use memcpy. If you cannot guarantee that the buffers don't overlap, you should use memmove.