They are the same garment: a knitted top you pull on over your head. Sweater is standard American English, jumper is standard British English, so choice usually comes down to which side of the Atlantic you are writing for.
The core difference
Sweater and jumper describe the same knitted, pulled-over garment, but sweater is American usage and jumper is British usage.
- sweater — a knitted top pulled on over the head, the usual American term: She wore a thick wool sweater on the chilly morning walk.
- jumper — the British word for the same garment: She pulled on a warm woollen jumper before heading out. In American English, jumper instead means a sleeveless dress worn over a blouse.
Why the same garment has two names
The confusion mostly comes from jumper carrying a second, unrelated meaning depending on where you are. In Britain, a jumper is simply a sweater. In the United States, a jumper is a sleeveless dress worn over a blouse, so an American reader picturing a jumper will imagine something quite different from a British one.
A jumper can also mean a short wire used to bridge a connection on an electrical circuit, which has nothing to do with clothing at all. If in doubt about your audience, sweater is the safer, unambiguous choice internationally.
