Neighbor is preferred in American English, and neighbour is preferred in all the other main varieties of English. The words are the same in every other respect. The spelling difference extends to derivative words such as neighborhood/neighbourhood, neighborly/neighbourly, neighbored/neighboured, and neighboring/neighbouring.
It’s neighbor vs. neighbor on rural Ritchie Highway [Baltimore Sun]
In summer, the rink area is a sprawling lawn, neighbored by restaurants and chess players. [LA Times]
The two neighborhoods have begun organizing a Marco Polo Day and an East Meets West Christmas Parade. [NY Times]
Outside the U.S.
Revitalizing historic neighbourhoods reduces the need to develop new land and costly new municipal infrastructure, a key objective of smart growth. [The Star Phoenix (Canada)]
A military crane parked in an Oxford street has attracted a barrage of criticism from angry neighbours. [Oxford Mail]
After helping ferry his neighbour’s children to safety, he returned to find a group of teenagers inside another house. [Brisbane Times]