Clamor and clamour mean exactly the same thing: a loud outcry, or to demand something noisily. The only difference is spelling — clamor is American English, clamour is British English. Think of the u as the British extra, the same pattern as colour and favour.
The core difference
They are the same word with the same meaning; clamor is the American spelling and clamour is the British spelling.
- clamor — American spelling: There was a public clamor for the minister to resign.
- clamour — British spelling: Residents clamoured for better bus services.
How to tell them apart
There is nothing to tell apart in meaning, only in spelling. Both words trace back to the Latin clamor, meaning a shout or cry, and both can work as a noun (the clamor/clamour of the crowd) or a verb (fans clamor/clamour for more).
The mix-up usually shows up when a writer switches between British and American house styles, or copies text from one and pastes it into the other. Pick one spelling and use it consistently throughout a document; UK publications and most of the world outside the US favour clamour, while US publications use clamor.