Looking for an explanation based on their different use cases.
Although their colloquial meaning is quite similar (to save ≈ to store), in git they're two completely unrelated sub-commands of the stash command. This is not really a comparison, but in short here's what they are :
stash save creates a stash entry, but is deprecated. There are no real use-cases not covered by stash push, which is implied when no other sub-command is given, so most people just do git stash to create a entry
stash store is a plumbing command to explicitly put a stash entry in the reflog (which is automatically done behind the scenes anyway, this is the way to explicitly fire it in a scripting context)