cease: it's much more formal, it works as an intransitive and only sometimes as a transitive, and it requires a different verb conjugation. eg: "the sun ceased to shine", "after that, her crying ceased" and "then she ceased her attacks on the king" are okay, but "i ceased his talking" isn't quite right. stop: less formal, but much more versatile and can be used transitively and intransitively. eg: "the sun stopped shining", "she stopped crying", "her crying stopped" and "then she stopped attacking the king" are okay, as well as "i stopped him from talking".