country vs British

country and British both are nouns.

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

As nouns, British is a hyponym of country; that is, British is a word with a more specific, narrower meaning than country:

country (noun)	British (noun)
a politically organized body of people under a single government	the people of Great Britain
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)

country (adjective)	British (adjective)
	of or relating to or characteristic of Great Britain or its people or culture