Commonwealth of Australia vs British Commonwealth

Commonwealth of Australia and British Commonwealth both are nouns.

Word	Noun	Adjective	Verb	Adverb
Commonwealth of Australia	Yes	No	No	No
British Commonwealth	Yes	No	No	No

Commonwealth of Australia (noun)	British Commonwealth (noun)
a nation occupying the whole of the Australian continent; Aboriginal tribes are thought to have migrated from southeastern Asia 20,000 years ago; first Europeans were British convicts sent there as a penal colony	an association of nations consisting of the United Kingdom and several former British colonies that are now sovereign states but still pay allegiance to the British Crown