The extra options supported by Win32
MoveFileEx function provide a good overview and explanation of the similarities and differences between renaming and moving.
When moving files across volumes, a copy (followed by delete) is performed.
From the perspective of an ordinary computer user, when one wants to perform a rename, one would expect the computer to prevent this from overwriting an existing file of the same name.
Whereas copying and moving would offer a choice of cancelling, auto-renaming, or replacing (overwriting).
Note that this is from a user's perspective, not from a programmer's perspective. However, the API must support additional flags so that the programmer can implement the user's expectations.
In summary, due to conventions inherited from the days of MS-DOS, a "rename" is perceived to be less destructive (other than being renamed) than a "move", even though the distinction no longer has any relevance to the underlying mechanism (the file system) unless the operation spans different file systems.
Unix and Linux had been designed by programmers, for programmers. Therefore the opinions of ordinary computer users had never factored into the design of file system commands, so the distinction does not exist there.