The difference between neighbor and neighbour are that one is the American way of spelling, and the other is the British way of spelling. Americans write neighbor, whereas the British write neighbor. The dictionary defines the word neighbor as someone who lives next to or near another person. The only difference is the way that they are spelled. The neighbor is the spelling used in English.
The main credit of changing the American spelling goes to Noah Webster, who wrote the first American dictionary in 1828. He was a firm believer that the spelling of words should be more like the way that they sound. Thus, the American spelling of "neighbor," as opposed to "neighbor."