football league vs conference

football league and conference both are nouns.

Word	Noun	Adjective	Verb	Adverb
football league	Yes	No	No	No
conference	Yes	No	No	No

As nouns, conference is a hypernym of football league; that is, conference is a word with a broader meaning than football league:

football league: a league of football teams
conference: an association of sports teams that organizes matches for its members

Other hypernyms of football league include league.

football league (noun)	conference (noun)
a league of football teams	a discussion among participants who have an agreed (serious) topic
	a prearranged meeting for consultation or exchange of information or discussion (especially one with a formal agenda)
	an association of sports teams that organizes matches for its members