What is the difference between act and rule utilitarianism?

The fundamental difference between act and rule utilitarian theories is that they have different criteria of right action and therefore prescribe different moral actions in different situations. A criterion of right action is the necessary condition(s) for an act to be right or moral. For Act Utilitarianism, the criterion of right action is as follows: An action A is right, if and only if it maximises utility. For Rule Utilitarianism, the criterion of right action is as follows: An action A is right, if and only if it is prescribed by a set of optimal rules which maximise utility if always applied.We can illustrate both the difference between the two systems, and one of the motivations behind rule utilitarianism with the following famous example. A doctor has five patients who are going to die without a transplant. The doctor has one other option - to kill and distribute the organs of a certain healthy and innocent individual to save the five people in hospital. There are two outcomes, one in which five people die, and one in which one person dies. Assuming their lives are all worth the same, an act utilitarian would say that as the death of one person is utility maximising compared with the death of five, therefore the doctor should kill the innocent person. This is a highly unintuitive implication of act-utilitarianism. A rule utilitarian however can prescribe the intuitive result that the doctor should not murder the innocent man. They claim that if we are to adhere to a set of moral rules, the set of moral rules which would produce the greatest utility contains the rule: do not murder someone innocent. Therefore, for a rule utilitarian, the doctor should not kill the innocent person to save the other patients, as this does not adhere to a set of optimal moral rules.