An argumentative essay tends to be a longer essay involving independent research, and aims to make an original argument about a topic. Its thesis statement makes a contentious claim that must be supported in an objective, evidence-based way.
An expository essay also aims to be objective, but it doesn’t have to make an original argument. Rather, it aims to explain something (e.g., a process or idea) in a clear, concise way. Expository essays are often shorter assignments and rely less on research.