rain vs rain shower

rain and rain shower both are nouns.

Word	Noun	Adjective	Verb	Adverb
rain	Yes	No	Yes	No
rain shower	Yes	No	No	No

As nouns, rain shower is a hyponym of rain; that is, rain shower is a word with a more specific, narrower meaning than rain:

rain (noun)	rain shower (noun)
anything happening rapidly or in quick successive	a brief period of precipitation
water falling in drops from vapor condensed in the atmosphere
drops of fresh water that fall as precipitation from clouds

rain (verb)	rain shower (verb)
precipitate as rain