Odor and odour mean exactly the same thing; only the spelling changes with the country. Odor is standard American English, odour is standard British English. A quick memory hook: odour keeps the u the same way colour and favour do in British spelling.
The core difference
Odor and odour are the same word for a smell, spelled the American way and the British way.
- odor — the American spelling, for a smell, pleasant or not: A strong odor of garlic filled the kitchen.
- odour — the British spelling, used the same way: An odour of damp hung over the old cottage.
How to tell them apart
There is no difference in meaning between odor and odour; both come from the Latin odor, meaning smell. The word covers any smell, good or bad, though it is used more often for unpleasant ones.
The mix-up mostly shows up in product labelling and cleaning or beauty copy, where American brands write odor and British ones write odour. Keep the spelling consistent with your intended audience across a single page or document.