The differences between an expository and an argumentative essay:

The purpose of exposition is to inform; the purpose of argumentation, on the other hand, is to convince.

 One essential characteristic of argument is the sense of opposition.  The concept of Argument has 2 sides to it--- proposition and opposition. What counts in an argumentative essay, then, is the writer's ability to create a sense of interior debate, of allowing others voices their say, and maintaining equilibrium among those voices. It's a matter of fairness and reasonableness.

An argumentative essay can employ expository techniques (e.g. defining, comparing, describing) in order to state, illuminate or support the presented arguments or counter-arguments.