Snacks and food of all kinds is allowed. The only things restricted are liquids, you can only have liquids in very small amounts when you go through security (check you airline for the amount allowed.)
Once you get through security you can buy any food or drink you want and bring it on the plane. We have gotten fast food, drinks and all, AFTER security and taken them on the plane to eat. Lots of people do this when traveling during a meal time.
The rules change all the time, so check before you go, but snacks have never been restricted.