"Brittany (French: Bretagne [bʁə.taɲ] ( listen); Breton: Breizh, pronounced [brɛjs] or [brɛχ]; Gallo: Bertaèyn, pronounced [bəʁ.taɛɲ]) is a cultural region in the north-west of France. Covering the western part of Armorica, as it was known during the period of Roman occupation, Brittany subsequently became an independent kingdom and then a duchy before being united to the Kingdom of France in 1532 as a province. Brittany has also been referred to as Less, Lesser or Little Britain (as opposed to Great Britain). It is bordered by the English Channel to the north, the Celtic Sea and the Atlantic Ocean to the west, and the Bay of Biscay to the south. Its land area is 34,023 km² (13,136 sq mi)."
So at some stage, Great Britain was used to distinguish it from Brittany (Little Britain). Great Britain is the largest island in the archipelago that is the British Isles. The British Isles comprises the island of Great Britain, the island of Ireland and all the smaller islands that make up the archipelago. England, Scotland and Wales make up Great Britain. As it is no longer strictly necessary to use "Great" to distinguish it from Brittany, Great Britain is often colloquially referred to as Britain.
The [United Kingdom]4 is officially the United Kingdom of Great Britain and Northern Ireland. The rest of the island of Ireland is the Republic of Ireland.
The Kingdom of Great Britain was formed out of the union of the English and Scottish crowns in 1707. Geographically, it comprised of the island of Great Britain and the smaller islands surrounding it (excluding the island of Ireland). When the Irish crown was merged with that of the kingdom in 1801, the United Kingdom of Great Britain and Ireland was formed. When the Irish Free State (now the Republic of Ireland) seceded from the United Kingdom in 1922, what was left was the United Kingdom of Great Britain and Northern Ireland, as it still is today.
Britannia was the name given by the Romans to the part of Great Britain over which they had some degree of control. That is roughly modern day England and Wales. The remaining part of Great Britain outside of Roman control was Caledonia (roughly modern day Scotland).
Now back to the original question. British and Briton are derived from the name the Romans used to refer to the originally Celtic peoples of Britain. The term British in particular is nowadays used colloquially to refer to anything or anyone from Great Britain, or even the UK.
The people of Brittany are referred to as Bretons. It would be almost as odd to call the people of Great Britain "Great British" as it would to call Bretons "Little British".
PS. I can't add more than 2 links, so I have moved all the links to my blog.