I guess these words are interchangeable in most cases ....
A "Reply" is usually a reaction to something in speech or writing, though it can also be a reaction of a different kind, like a gesture or an action.
In general, it seems to me that a "reply" is a more cognitive thing that you'll normally receive from a person.
ex- "reply" to a letter; He didn't reply to her greeting.
"Answer" means to speak or write in reply. It implies the satisfying of a question, demand, call, or need .
ex- I asked him a simple question and he gave me a long and confusing answer.
"answer a phone call" - I don't think we "reply" to a phone call.