Irish vs Gaelic

Irish and Gaelic both are nouns.

Word	Noun	Adjective	Verb	Adverb
Irish	Yes	Yes	No	No
Gaelic	Yes	Yes	No	No

As nouns, Gaelic is a hypernym of Irish; that is, Gaelic is a word with a broader meaning than Irish:

Irish: the Celtic language of Ireland
Gaelic: any of several related languages of the Celts in Ireland and Scotland

Other hypernyms of Irish include Erse, Erse language, Gaelic language, Goidelic, Goidelic language.

Irish (noun)	Gaelic (noun)
the Celtic language of Ireland	any of several related languages of the Celts in Ireland and Scotland
whiskey made in Ireland chiefly from barley
people of Ireland or of Irish extraction

Irish (adjective)	Gaelic (adjective)
of or relating to or characteristic of Ireland or its people	relating to or characteristic of the Celts