theater vs music hall

theater and music hall both are nouns.

Word	Noun	Adjective	Verb	Adverb
theater	Yes	No	No	No
music hall	Yes	No	No	No

As nouns, music hall is a hyponym of theater; that is, music hall is a word with a more specific, narrower meaning than theater:

theater: a building where theatrical performances or motion-picture shows can be presented
music hall: a theater in which vaudeville is staged

theater (noun)	music hall (noun)
a building where theatrical performances or motion-picture shows can be presented	a theater in which vaudeville is staged
the art of writing and producing plays	a variety show with songs and comic acts etc.
a region in which active military operations are in progress