Harbour vs. Harbor
It might seem difficult to make a difference between "harbour" and "harbor", but it shouldn't at all. Everything here is about the different spellings preferred by American and British English, or more exactly by US and UK English.
More exactly, it's not the meaning of your message, or the meanings of "harbour" and "harbor" those you should worry about. These are the same. The matter here is actually the right context where you should use each. So when do we use "harbour" and "harbor" and why?
Harbour vs. Harbor
Both words carry the same definitions, both as verbs and as nouns. As nouns, they are defined as an area near a coast, where ships are kept away from the water of the sea. As verbs, "harbour" and "harbor" can either refer to feeling or thinking about something for a long while (e.g.: "to harbor/harbour doubts/feelings/thoughts/hopes), or to hiding/covering something or someone bad or wrong (e.g.: to habour/harbor a murderer/murder/criminal/crime etc.).
The only difference illustrated by their different spellings is the type of English where they are used. "Harbour" is the UK spelling, whereas "harbor" is the US spelling.
When do we use "harbour"?
"Harbour" is the UK spelling for the noun/verb with the same meaning as "harbor". So even though you can replace it in any context where "harbor" is used, it is recommended that you spell "harbour" especially if you are engaged in a British conversation.
When do we use "harbor"?
The same applies as above. "Harbor" is a perfect synonym for "harbour", though it's recommended you spell it without "u" if you have a conversation in US English, whether it's American, Australian etc.
There is no remarkable difference between "harbour" and "harbor", apart from the spelling preferred by UK and US English. "Harbour" is the version preferred by the British, while Americans will recommend "harbor". Apart from these linguistic subtleties, dictionaries do not identify any notable difference between the meanings of these words, both as nouns and as verbs.