Difference between For pity’s sake and Out of pity

for pity’s sake—is used when pleading for smth.:

• When you drive, for pity’s sake don’t smoke with one hand while you’re trying to turn a corner.

out of pity—do smth. because of a feeling of pity:

• If I continue this relationship, I’d be doing it out of pity not actual love and that isn’t fair for her either.