baton vs billy club

baton and billy club both are nouns.

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

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

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

baton (noun)	billy club (noun)
a thin tapered rod used by a conductor to lead an orchestra or choir	a short stout club used primarily by policemen
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