In ordinary situations I see no difference in meaning, and I believe they would be used interchangeably. I have a slight stylistic preference for the former. "the British defence minister" could in theory mean that he is British, and is a defence minister, but not a defence minister of Britain.
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AnonymousWhat's the difference between ''Britain's defence minister'' and the British defence ministerTechnically, Britain's is a complement. The defence minister is the defence minister of Britain.