Let’s start with the basics. To think of these hats is to consider each as a family. There’s a cowboy hat family and fedora family. Unfortunately, though there’s some gross crossbreeding and the waters get murky in this hat utopia. But if we take a peek at a hat’s DNA, it derives from some origin and can usually be classified one way or another.
As a rule, a cowboy hat will have its brim turned up on both sides. Conversely, a fedora will have the back brim down with the front turned up.