When should you send a thank-you letter after an interview? Simple … always!
Never mind the circumstance, your interest, whatever, always send a thank-you letter, it is simply good manners. It can also give you an advantage in a horribly competitive employment market. Sending a thank-you letter puts your name in front of the employer (again), it confirms that you know how to behave, that you know how to deal with people, that you follow through on tasks and have a determination to succeed.
Some years ago the thank-you letter was no more than a polite formality; today, you should view this item as another opportunity to reinforce your application. Highlight information about yourself that is particularly relevant to the job requirements; confirm that you are capable of meeting their needs, provide examples of previous experience that will be of benefit to them, reinforce your enthusiasm for the opportunity. Correctly formulated, a thank-you letter can differentiate you from other candidates, particularly those who did not bother to write one.