Some expressions become "frozen", and this is one: we usually say that God answers prayers. In this context, it usually means that God has given a favourable answer, usually by giving the person praying whatever he or she asked for. (There's nothing in the verb "answer" that suggests this; it is simply what people usually mean when they say that God answered their prayers.)
"Responded" here simply means that God did something in response to the prayer, but it's not clear what. For all we know, he could have responded by striking the petitioner with a lightning bolt.
If God replies to a prayer, it means he responds by saying something in return. Of course, if you believe God does speak to people in this way, that might be exactly what you mean; but usually, God is thought to respond in a less verbal manner, perhaps by curing somebody of cancer or something.
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