Apathy is not caring; Stoicism is caring deeply but refusing to be ruled by what you can't control. A Stoic works on the judgement underneath the feeling rather than numbing it — an apathetic person just checks out.
Side by side
Aspect
Stoicism
Apathy
Feeling
Worked at the judgement
Numbed / absent
Caring
Cares deeply
Doesn't care
Action
Engaged, deliberate
Withdrawn
Aim
Flourishing
None
What each one means
Stoicism
In ordinary English, uncomplaining endurance of hardship without showing what it costs. In philosophy, a school founded in Athens and carried on in Rome by Seneca, Epictetus and Marcus Aurelius: virtue is the only real good, we should live in agreement with nature and reason, and peace comes from telling what is in our control from what is not.