ballet vs modern ballet

ballet and modern ballet both are nouns.

Word	Noun	Adjective	Verb	Adverb
ballet	Yes	No	No	No
modern ballet	Yes	No	No	No

As nouns, modern ballet is a hyponym of ballet; that is, modern ballet is a word with a more specific, narrower meaning than ballet:

ballet: a theatrical representation of a story that is performed to music by trained dancers
modern ballet: a style of ballet that admits a wider variety of movements

Other hyponyms of ballet include classical ballet, comedy ballet.

ballet (noun)	modern ballet (noun)
a theatrical representation of a story that is performed to music by trained dancers	a style of ballet that admits a wider variety of movements
music written for a ballet