They mean exactly the same thing. As is often the case, there isn't a perfectly sharp distinction between US and British English. In the US, "period" is the usual term. "Full stop" is understood.
If a US speaker happens to use "full stop," it doesn't mean anything in particular.
I notice that the American Heritage Dictionary's definition of "period" is "A punctuation mark ( . ) indicating a full stop..."
As a side note, over some period of time, perhaps 1900 to 1960, telegrams were printed entirely in uppercase without punctuation, and the end of a sentence was indicated by the word "STOP." Since the price of a telegram was billed by the word, it let to a style of ungrammatical, abbreviated English called "telegraphic" English. An example, from a novel:
"ARTHUR STOP COME AT ONCE STOP AM IN TERRIBLE TROUBLE STOP DO NOT PHONE STOP AUNT MATILDA"