From a purely semantic viewpoint, "awase" comes from the verb "awaseru" which, in turn, comes from the verb "au" (all of which use the same character as "ai" in "aikido"). "Awaseru" basically means "to match <something> up with <something else>" and "awase" is basically its noun form.
"Musubi" comes from the verb "musubu" which basically means "to tie."
For me, at least, in an aikido sense, "awase" carries the connotation of matching one's movements up with someone else's. I can't say I've used that term very much where I've trained. We use "musubi" to connote a sense of connection, whether physical or spiritual, with your partner...