In addition to the points Naomi makes, here are some other differences:
1. In some contexts, “answer” means to attempt to provide information that is being asked for. In contrast, a response or a reply may simply be “I don’t know the answer” or some other response that is not an attempt to answer the question. Example: “The teacher asked three students ‘What is the square root of 49? One student answered correctly, and the second student answered the question incorrectly. The third student did not even try to answer, but simply said “I don’t care about math.’”
2. “Reply” is always verbal, but “respond” can be non-verbal. Example: “He didn’t reply to my statement. He just responded with a shrug of his shoulders.”