Both Epicureanism and Stoicism were Hellenistic philosophies that centered around how people should live their lives. Both of them taught people to try to avoid excessive desires and passions. Both of them taught that people should not fear death. These are the major similarities between the two.
The two differed to some extent in their prescriptions for how to avoid excessive desires and passions. Epicureans believed that people should withdraw themselves from the world. They should live with their friends but try to keep away from the cares of the everyday world so that they could remain tranquil. By contrast, the Stoics felt that people should remain in society and should simply take what happened to them in the manner that we now call "stoic." They should achieve tranquility by being tough and controlling themselves, not by running away from the problems as the Epicureans advocated.