climb up vs climb

climb up is not a noun while climb is a noun.

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

As verbs, climb up and climb are synonyms defined as:

Other synonyms of climb up include go up, mount.

climb up (noun)	climb (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 up (verb)	climb (verb)
rise in rank or status	go up or advance
go upward with gradual or continuous progress	increase in value or to a higher point
appear to be moving upward, as by means of tendrils	improve one's social status
	go upward with gradual or continuous progress
	move with effort, by grasping
	slope upward