The term Great Britain, abbreviated to Britain, should not be used interchangeably with the United Kingdom, or UK, because they are different. Here's how.
Great Britain and the UK
Great Britain consists of the countries of England, Scotland and Wales, which together form the mainland of the British Isles.
The United Kingdom comprises Great Britain plus the province of Northern Ireland, on the island of Ireland, bordering the Republic of Ireland (a separate country and a member of the European Union) which means Britain is only a part of the UK, not the whole. Here is a helpful diagram showing the relationships between these countries and terms.
The United Kingdom is a country in the way that most people think of countries, for example the USA, Spain and India.
UK membership of international organizations
Use of term British to indicate the whole of the UK
However, some national and international endeavours are organized under the term Great Britain, in particular in sport, where it is Team GB that competes in the Olympics (despite being funded by the organization UK Sport) and many national sports governing bodies, such as British Cycling and British Swimming, use the terms Britain, Great Britain or British.
The currency of the UK is known as the British Pound (GBP), even though it is the currency of the whole of the United Kingdom.
The British Armed Forces are also responsible for the defence of the whole of the United Kingdom.
The term British also applies to all citizens of the United Kingdom, regardless of where they live. Even residents of Northern Ireland are British, although those with nationalist sentiment there may refer to themselves as being Irish, or Northern Irish. A 2018 report in the Belfast Telegraph newspaper found that 46.7% of people there felt British, 58.6% felt Irish and 57.9% felt Northern Irish, with many people feeling affinity to multiple identities.
Isle of Man - a British Crown Dependency
It is worth noting that the British Crown Dependencies of the Isle of Man (an island between England and Northern Ireland) and the Channel Islands (the small islands of Jersey and Guernsey just off the coast of France) are not part of the United Kingdom, with separate parliament and laws even though their foreign affairs and defence matters are managed by the United Kingdom government. The Isle of Man has never been part of the UK or of the European Union.
The currencies of these territories are different. The Isle of Man uses the Manx Pound, but it is maintained at parity with GBP. It is the same with the Jersey Pound and Guernsey Pound.