climb up vs come up

climb up and come up both are verbs.

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

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

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

climb up (verb)	come up (verb)
rise in rank or status	result or issue
go upward with gradual or continuous progress	start running, functioning, or operating
appear to be moving upward, as by means of tendrils	gather or bring together
	gather (money or other resources) together over time
	bring forth, usually something desirable
	come up, of celestial bodies
	come to the surface
	move toward, travel toward something or somebody or approach something or somebody
	move upward
	get something or somebody for a specific purpose
	originate or come into being
	be mentioned