Irish Republic vs Republic of Ireland

Irish Republic and Republic of Ireland both are nouns.

Word	Noun	Adjective	Verb	Adverb
Irish Republic	Yes	No	No	No
Republic of Ireland	Yes	No	No	No

As nouns, Irish Republic and Republic of Ireland are synonyms defined as:

Irish Republic and Republic of Ireland: a republic consisting of 26 of 32 counties comprising the island of Ireland; achieved independence from the United Kingdom in 1921

Other synonyms of Irish Republic include Eire, Ireland.

Irish Republic (noun)	Republic of Ireland (noun)
a republic consisting of 26 of 32 counties comprising the island of Ireland; achieved independence from the United Kingdom in 1921	a republic consisting of 26 of 32 counties comprising the island of Ireland; achieved independence from the United Kingdom in 1921