Both are often used interchangeably. An answer is given in response to a question. A reply can be the same an an answer, or it can be a response to a statement or a comment, so it has a broader scope. Respond is basically the same as reply, but has an even broader scope as it can include nonverbal as well as the verbal and written element.

Most of the time, answer, reply, and respond can be used interchangeably; they are synonyms. But you are right, an answer is always a response to a question. Reply and respond can be used to describe conversation even when a question was not asked.

Answer, reply and respond are very close. You can answer a question,but for the verbs reply and respond you need to add the word "to" He responded, replied TO the question.