None whatsoever. They are exactly the same word. 'Rumour' and 'rumor' are simply alternative spellings for the same word, and the spelling makes no difference to either the meaning or the pronunciation. Other words which have the similar alternative spellings are colour/color and humour/humor. 'Rumour' is the original spelling, used everywhere apart from the USA, and to a certain extent in Canada. In the early 19th century, an American spelling reformer called Noah Webster decided to simplify English spelling and removed the 'u' from these words. So while the US writes this word 'rumor', without the 'u', Britain, Ireland, Australia, NZ, S. Africa and more conservative Canadians use the original spelling with a 'u'.