For one thing, "polite" is an adjective, and "courtesy" is a noun. "The woman was very polite. She did not offend anyone." The two words are similar, but "courtesy" has some special meanings. See this entry: http://www.learnersdictionary.com/search/courtesy

Polite- 1. showing regard for others, in manners, speech, behaviour, etc.; courteous 2. cultivated or refined 3. elegant or polished Courtesy- 1. Polite behavior 2. A polite gesture or remark a. Consent or agreement in spite of fact; indulgence: b. Willingness or generosity in providing something needed: adj. 1. Given or done as a polite gesture: I paid a courtesy visit to the new neighbors. 2. Free of charge: www.Thefreedictionary.com