What's the difference, if any, between someone who says they are British and someone who says they are a Briton?

Most of the time, "British" and "a Briton" mean exactly the same thing-- a person from England, Wales, or Scotland, which together make up Great Britain.

However, "Briton" has a couple of other possible (but much less common) meanings. It could mean a member of the ancient tribe, the Britons, who occupied part of Great Britain before they were displaced by the Romans and by other tribes. Great Britain is named for the Britons.

Or it could be a variant spelling of "Breton," which is the name for a person of Bretagne (in northwest France). The Bretons are people of Celtic heritage and language, culturally related to the Welsh and Cornish peoples of Britain.

Boy what a stupid question.

Spelling

There is no difference

None. Though a Welshman would be more likely to use the word Briton, it's only a regional dialect thing.

Briefly, a "Briton" can have several different meanings. They were a Celtic people who occupied the British Isles at the time of the Roman invasion. When the Romans withdrew the Anglo-Saxons (English) invaded and occupied what is today England. Many Britons were displaced and ended up as refugees on the coast of France in what is today known as Brittany. Over time the English consolidated their hold over all of the British Isles, and the various ethnic groups who live there are called British. So, a Briton might be someone from the British Isles, might be someone from Brittany, or might refer to a Roman era native of the British Isles.