I think the difference is the degree of intensity.

Stress is mental tension (stuck in traffic, have an appointment in 2 minutes).

Anxiety is fear , uneasiness, worry about something real or not real (anxiety about driving a car because of a fear of being in an accident which may or may not happen).

Stress keeps us on our toes. Stress makes us be on time.
Anxiety keeps us from being able to do what we want to do. Anxiety prevents us from even showing up!