If someone looks careworn, their face looks sad, worried, and tired — as if worry has been wearing them down for a long time. You use it about how a person looks, not about a passing bad mood. A writer might speak of careworn eyes or a careworn mother. It is a bookish, gentle word; friends in a café would more often say tired or stressed. Though actually the care here is the old meaning “sorrow, worry”, not “looking after someone”. A careworn person is worn by trouble, not worn out from giving care — even if those two lives sometimes meet.