majors vs big league

majors and big league both are nouns.

Word	Noun	Adjective	Verb	Adverb
majors	Yes	No	No	No
big league	Yes	No	No	No

As nouns, majors and big league are synonyms defined as:

majors and big league: the most important league in any sport (especially baseball)

Other synonyms of majors include major league.

majors (noun)	big league (noun)
the most important league in any sport (especially baseball)	the most important league in any sport (especially baseball)