“Brittonic languages” is a more recent coinage (first attested 1923 according to the Oxford English Dictionary) intended to refer to the ancient Britons specifically.
In English, the term “Briton” originally denoted the ancient Britons and their descendants, most particularly the Welsh, who were seen as heirs to the ancient British people.
After the Acts of Union 1707, the terms British and Briton came to be applied to all inhabitants of the Kingdom of Great Britain and its empire.
With the beginning of Anglo-Saxon settlement and Gaelic Scots in the 5th century, the culture and language of the Britons fragmented and much of their territory was taken over by the Anglo-Saxons and Scots Gaels.
The extent to which this cultural and linguistic change was accompanied by wholesale changes in the population is still a matter of discussion.
Common Brittonic developed into the distinct Brittonic languages: Welsh, Cumbric, Cornish and Breton.
The earliest known reference to the inhabitants of Britain seems to come from 4th century BC records of the voyage of Pytheas, a Greek geographer who made a voyage of exploration around the British Isles between 330 and 320 BC.
During this period some Britons migrated to mainland Europe and established significant settlements in Brittany (now part of France) as well as Britonia in modern Galicia, Spain.