climb vs move up

climb is a noun but move up is not a noun.

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

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

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

climb (noun)	move up (noun)
the act of climbing something
an event that involves rising to a higher point (as in altitude or temperature or intensity etc.)
an upward slope or grade (as in a road)

climb (verb)	move up (verb)
go up or advance	move upward
increase in value or to a higher point	move to a better position in life or to a better job
improve one's social status
go upward with gradual or continuous progress
move with effort, by grasping
slope upward