tempo vs rate

tempo and rate both are nouns.

Word	Noun	Adjective	Verb	Adverb
tempo	Yes	No	No	No
rate	Yes	No	Yes	No

As nouns, rate is a hypernym of tempo; that is, rate is a word with a broader meaning than tempo:

tempo (noun)	rate (noun)
the rate of some repeating event	the relative speed of progress or change
(music) the speed at which a composition is to be played	amount of a charge or payment relative to some basis
	a quantity or amount or measure considered as a proportion of another quantity or amount or measure
	a magnitude or frequency relative to a time unit

tempo (verb)	rate (verb)
	assign a rank or rating to
	estimate the value of
	be worthy of or have a certain rating