The argumentative essay is written in a factual manner with the information tending to resemble one side of a debate. In this style of essay, you present statistics and provable information to the reader in a formal, even academic, manner.
In a persuasive essay, you present your opinion to the reader, which may or may not be bolstered with facts. This style is more personal, less formal, perhaps even written in a conversational style--depending on the topic.
Those are the differences. The similarity is that both these types of essay are written to influence the reader.