"Rainy" is an adjective. It describes a time period, usually, like "a rainy day" or "a rainy afternoon."
"Rain" is a noun or verb.
As a noun, it refers to the actual water drops that fall from the sky, as in "The rain fell on my head." Or " They rain made puddled on the ground."
As a verb, "rain" means the action of the water falling. "It is raining today."