come down vs rain down

come down and rain down both are verbs.

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

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

Other hyponyms of come down include rain, spat, snow, hail, sleet.

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