First of all, they're different parts of speech. Courtesy/Politeness-Noun Courteous/Polite-Adjective Courteously/Politely-Adverb That being said, in general they're interchangeable. As Jackie said, courtesy has an extra meaning only used in business transactions. Courteous is a little more formal. If I wanted to split hairs, I might say that "courteous" implies a certain cultural norm of being nice (holding the door, looking someone in the eyes when you speak with them, pulling out a chair for a lady at dinner, etc.) whereas "polite" is less specific and just means "don't be a jerk" (e.g. don't mention how someone's house smells bad). But for all intents and purposes they just mean "really nice".

The Oxford dictionary defines polite as “having or showing behaviour that is respectful and considerate of other people.” And it defines courtesy as “the showing of politeness in one’s attitude and behaviour towards others.” in practical use there is a slight difference in tone where courtesy is more heartfelt warmth being expressed. The slight difference is when one shows politeness which is common or standard practice in human interaction (as expected from cultural environment) but showing courtesy is more emotionally attached and welcoming as well as more considerate to the recipient.

I would say in American English there is no real difference between the two in almost all circumstances. One difference is that courtesy is sometimes used to mean free/no charge. As an example, if you are having repairs done on your car, you might be given a "courtesy car" by the repair shop to drive while your car is worked on.