As a non-native speaker, the word “courteous” is not a word I encounter often and it’s not a word I have ever used myself. I’m not sure myself exactly what the difference is in the actual meaning of those two words. I have never heard the word “courteous” used by any native speaker of English I have known. I imagine the reason why I’m the first one to answer your question is that most native speakers don’t really know what the difference is either.
“Courteous” literally means “having manners fit for a royal court”.
Society has changed. There was a time, when there was a very clear difference between different classes of society. People from the aristocracy lived in a world that was completely different from the world a farmer, for instance, lived in. For instance, 200 years ago a woman from the aristocracy had to learn the correct way to enter and exit a carriage (a horse drawn vehicle). Some of the words that were used in the context of this very different social order have survived, but don’t make much sense any longer.