Перевод - Турецкий-Английский - Bana cevap yazmayacaksın mısın?

You should use one of the followings:

"to respond", "to write back", "to write an answer"

"wont you reply me?"

Agree with minuet

Better would be:
" Won't you write me an answer?"

Dear friends,
Do you see a big difference in the literal sense ?

For me yes, to use "answer" changes meaning.
we use that verb for telephone calls generally.

However, "reply" can be used for letters, mails etc.

I agree with you all girls.

won't you reply me?