come up vs climb up

come up and climb up both are verbs.

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

As verbs, climb up is a hyponym of come up; that is, climb up is a word with a more specific, narrower meaning than come up:

come up (verb)	climb up (verb)
result or issue	rise in rank or status
start running, functioning, or operating	go upward with gradual or continuous progress
gather or bring together	appear to be moving upward, as by means of tendrils
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