These verbs have essentially the same meaning. You can:
When someone calls you, you answer the phone (or pick up the phone).
When you make a statement or some comments, we usually say the other person replies or responds.
When you ask a question, we usually say the other person answers. However, this is not 100% – reply and respond can also be used for answering a question.
When used as a noun (with a, an, the and possessive pronouns like my, your, his, her, etc.) the words answer and reply don’t change – but the noun form of “respond” is response.