Republic of Ireland vs Irish Republic

Republic of Ireland and Irish Republic both are nouns.

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

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

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

Other synonyms of Republic of Ireland include Eire, Ireland.

Republic of Ireland (noun)	Irish Republic (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