commonwealth country vs country

commonwealth country and country both are nouns.

Word	Noun	Adjective	Verb	Adverb
commonwealth country	Yes	No	No	No
country	Yes	No	No	No

As nouns, country is a hypernym of commonwealth country; that is, country is a word with a broader meaning than commonwealth country:

Other hypernyms of commonwealth country include body politic, commonwealth, land, nation, res publica, state.

commonwealth country (noun)	country (noun)
any of the countries in the British Commonwealth	a politically organized body of people under a single government
	the people who live in a nation or country
	the territory occupied by a nation
	an area outside of cities and towns
	a particular geographical region of indefinite boundary (usually serving some special purpose or distinguished by its people or culture or geography)