Both target tranquillity, from opposite metaphysics. Epicureans get there by removing fear of gods and death through atomism, and by withdrawing from public life. Stoics get there by aligning with a rational cosmos and staying in it. Same destination, opposite instructions about whether to show up to work.
Side by side
Aspect
Epicureanism
Stoicism
Route
Remove fear
Align with reason
Public life
Withdraw
Participate
Cosmos
Atoms and void
Rational order
Goal
Ataraxia
Ataraxia
What each one means
Epicureanism
Epicureanism is Epicurus's teaching that the good life is a quiet one — freedom from bodily pain and mental disturbance, reached by wanting less. In ordinary English the word also means devotion to refined pleasure, especially good food and drink — close to the opposite of what he taught.
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.