billy club vs baton

billy club and baton both are nouns.

Word	Noun	Adjective	Verb	Adverb
billy club	Yes	No	No	No
baton	Yes	No	No	No

As nouns, billy club and baton are synonyms defined as:

Other synonyms of billy club include billy, billystick, nightstick, truncheon.

billy club (noun)	baton (noun)
a short stout club used primarily by policemen	a thin tapered rod used by a conductor to lead an orchestra or choir
	a hollow cylinder passed from runner to runner in a relay race
	a hollow metal rod that is wielded or twirled by a drum major or drum majorette
	a short staff carried by some officials to symbolize an office or an authority
	a short stout club used primarily by policemen