both have the similar English of "answer," but when do you use one over the other?
- Anybody can ask a question
- Anybody can answer
- The best answers are voted up and rise to the top
But 答え has another meaning "問題・設問などを解いて出される結果", which what I would summarize as equivalent to "solution", while 返事 has "返答の手紙", which is equivalent to "reply".
I guess I could say that 答え is answer to a call/address or a question, while 返事 is answer to a call/address or a letter. Or put it another way, you can use 答え or 返事 interchangeably when referring to answering a call/address but for a solution to a question or problem 答え should be use, and for answering a letter 返事 should be used.