This is a great question, and one that many people have as all 3 desserts have a warm and bubbly fruity base. Really the only difference comes in the form of the topping.
Crumble – has a crumbly streusel like topping sprinkled over the top of the dish
Crisp – uses an oat in the streusel topping and sprinkled or crumbled over the dish
Cobbler – is topped with some kind of dough/biscuit/cake dropped over the top
There are no real hard rules though, so if you choose to add oatmeal to your streusel topping I don’t think anyone’s going to complain.