It's quite simple Rob.
Great Britain is the larger island of the archipelago that makes up the British (and Irish) Isles - usually taken to mean England, Scotland and Wales.
The United Kingdom is properly "The United Kingdom of Great Britain (see above) and Northern Ireland" but does not include the Crown Dependencies (Jersey, Guernsey, Isle of Man - which are all self-governing, but the UK conducts external affairs for - things like defence, consular service, etc)
The "British" team ("TeamGB") at the Olympics is representative of Great Britain and Northern Ireland, plus Crown Dependencies and all British Overseas Territories that do not have their own National Olympic Association (so the Falklands is part of Team GB, Bermuda is not because it has it's own NOA)