SMS stands for "Short Message Service", all text messages are SMS messages. So any SMS app will use your text messages, they are essentially just different versions of the default messaging app. There are however apps that allow you to send SMS messages over the internet for free. I don't use them so i can't recommend one really. They also generally don't let you keep your number either.
You could check out Google Voice though, which does essentially what you want, but it requires a new number. You get to choose your phone number though