This suggests that the person asking the question thinks that the other person is actually refusing to reply. If you "won't" do something, it means that you refuse to do it.
Why don't you reply to my text messages?
This suggests that the person asking the question simply wants to know why they don't receive replies to their text messages. There could be several explanations including a problem with the mobile phone, the contract provider problem or, of course, that the recipient is simply ignoring the messages.