"Cuocere" refers to the actual act of something transitioning from the state of being raw to the state of being cooked. For example, you would use cuocere in this context: "the hamburger meat cooked too fast on the high heat". Cucinare, on the other hand, is used to refer to the act of cooking or preparing a meal. Like, "I cooked breakfast earlier than usual this morning."
Cuocere: La carne deve cuocere a fuco lento. Il sole mi cuoce la pelle oggi. (Here it is used in a slightly figurative sense, that the sun is "cooking your skin" because it is so hot.
Cucinare: Mi piace cucinare più di qualsiasi altra cosa. // Ieri ho cucinato la cena per mia madre.