climb vs climb up

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

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

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

Other synonyms of climb include go up, mount.

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