Anytime a person returns communication it can be called a response or a reply, while an answer is a form of response which is a solution to a problem or question. So response and reply are generic and can be used in any situation, while answer is more specific in its usage.
So if you asked a question or asked for a solution to a problem, and the person gave it to you, then you can say "Thanks for your quick answer". If it was not in one of these categories, then use response or reply since these are both generic.
And if you are still in doubt, remember that because response is generic you can use it in any situation.