big league vs majors

big league and majors both are nouns.

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

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

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

Other synonyms of big league include major league.

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