The difference is very slight. Polite refers to what we would consider "basic manners". Courteous is what we would call "being a gentleman". The word courteous comes from the french "courtoisie" which referred to the manners that were appropriate in the king's court. In fact, courteous people are all polite but not all polite people are courteous. In common usage, the two words are used interchangeably.

Well, for me, being the gentleman I am...courteous might be opening the car door for a lady to get in, pulling out her chair for her to sit on at a restaurant or opening the restaurant door for her. Polite on the other hand might be~making "polite" conversation. Expressing an interest in the other persons interests without bringing up subjects which may be difficult, such as politics and religion. :-)