Britain is the actual name of the main island . So one would say Britain and Ireland meaning the two islands . You can also say the British Isles meaning Britain and the smaller islands that are close by . At the start of the 19th century, Ireland had been causing trouble with English rule, so the soldiers and colonists referred to Great Britain as the owner of the Empire, mostly to snub the Irish who certainly contributed to the Empire but whose loyalty was always a bit shaky .
In history, a tribe of celts was actually called the Britons, so you may see that some where .
The actual name Britain can be traced back to an ancient Greek Cartographer who referred to the place as Britain . This is interesting because the Jews and later the Phoenicians traded with the British Isles for tin, and the words Brit Ain are Hebrew for the land or people of the contract . Thus it may be the oldest word currently used in the British language .
An almost identical goddess Britannia to the one currently used can be found in ancient Hittite cities and subsequently on Phoenician coins . She was the Goddess of Waters .