mountain climber vs climber

mountain climber and climber both are nouns.

Word	Noun	Adjective	Verb	Adverb
mountain climber	Yes	No	No	No
climber	Yes	No	No	No

As nouns, climber is a hypernym of mountain climber; that is, climber is a word with a broader meaning than mountain climber:

mountain climber: someone who climbs mountains
climber: someone who climbs as a sport; especially someone who climbs mountains

Other hypernyms of mountain climber include adventurer, venturer.

mountain climber (noun)	climber (noun)
someone who climbs mountains	an iron spike attached to the shoe to prevent slipping on ice when walking or climbing
	someone who climbs as a sport; especially someone who climbs mountains
	someone seeking social prominence by obsequious behavior
	someone who ascends on foot
	a vine or climbing plant that readily grows up a support or over other plants