rain down vs come down

rain down and come down both are verbs.

Word	Noun	Adjective	Verb	Adverb
rain down	No	No	Yes	No
come down	No	No	Yes	No

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

Other hypernyms of rain down include fall, precipitate.

rain down (verb)	come down (verb)
precipitate as rain	get sick
	be the essential element
	criticize or reprimand harshly
	move downward and lower, but not necessarily all the way
	fall from clouds