climb up vs move up

climb up and move up both are verbs.

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

As verbs, move up is a hypernym of climb up; that is, move up is a word with a broader meaning than climb up:

Other hypernyms of climb up include arise, come up, go up, lift, rise, uprise.

climb up (verb)	move up (verb)
rise in rank or status	move upward
go upward with gradual or continuous progress	move to a better position in life or to a better job
appear to be moving upward, as by means of tendrils