move back vs back up

move back and back up both are verbs.

Word	Noun	Adjective	Verb	Adverb
move back	No	No	Yes	No
back up	No	No	Yes	No

As verbs, back up is a hyponym of move back; that is, back up is a word with a more specific, narrower meaning than move back:

Other hyponyms of move back include fall back, retreat, retrograde, back down, back off.

move back (verb)	back up (verb)
pull back or move away or backward	establish as valid or genuine
	become or cause to become obstructed
	make a copy of (a computer file) especially for storage in another place as a security copy
	move backwards from a certain position
	give moral or psychological support, aid, or courage to