The English version of words are spelt differently to the American English versions, most have a U, Neighbour, honour, etc...
Unfortunately most software has the American English version of the words, missing out the U, Neighbor, Honor, which has meant that you now have both options, in a dictionary, but the online Cambridge Dictionaries provide you with the option of British English, and American english, versions.
Thus you cannot claim that neighbourhood is spelt incorrectly, as both versions are accepted, it is merely the British English version rather than the American English version.
I hope this clears up any confusion.