In American English, maneuver is the correct spelling of the word meaning (among other things) a controlled change in movement or direction. Manoeuvre is the preferred spelling everywhere else. Maneuver and manoeuvre are pronounced the same, and they share all their definitions.
Examples
For example, these American publications use maneuver:
The flag requires 50 adults—or very strong teenagers—to maneuver the poles of the flag and hold them steady for the parade. [Chicago Sun-Times]
Then he opened the door, nimbly maneuvered his 6-foot-5 frame over his belted-in son and settled into the bucket seat between his children. [Wall Street Journal]
Mr. Chiluba cycled through vice presidents and cabinet members he suspected of maneuvering to take his power. [Washington Post]
And these non-U.S. publications use manoeuvre:
Prince William will help perform the daring manoeuvre of landing a helicopter on water during the royal visit to Canada. [Evening Standard]
Collingwood coach Mick Malthouse said he manoeuvred to make sure no one player copped too prolonged a buffeting. [Sydney Morning Herald]
Certainly Clark must see that this is obviously political manoeuvring to get her elected. [Vancouver Sun]
