There is no difference in meaning between harbor and harbour. Harbor is the preferred spelling in American English, and harbour is preferred in all other major varieties of English.
Examples
For example, these U.S. publications use harbor:
Deepening the Savannah harbor to accommodate larger ships is now the number one priority for officials in the coastal city. [GPB]
State Senator Jack Hart is advocating for the city to dump its snow surplus into the harbor, saying that Boston streets have become a public safety concern. [Boston Globe]
Baltimore’s harbor may be a mess, but those who attended a daylong conference on its problems Saturday came away encouraged that it doesn’t have to stay that way. [Baltimore Sun]
And these non-U.S. publications use harbour:
The city, with its excellent harbour, occupied far too important a strategic position to be left deserted for long. [The Guardian]
To many Sydneysiders, the harbour is a proudly flaunted jewel, a brilliant vision to admire from balconies, boats and foreshores. [Sydney Morning Herald]
Much of historic St. John’s centres around its harbour and the major downtown shopping area is just metres from the water. [Toronto Sun]
