“Engine” is used to describe an assembly of parts that contain a motor and the systems to make the motor more efficient and self-sustaining.
“Motor” is used to describe a device that converts energy into motion
Bringing up details from my comment below, the motor component of a traditional car engine is the piston/crank/cylinder assembly; as this is the part that converts fuel into motion. The rest of the engine is a necessary part to keeping the motor running.
However, when it comes to cars, people tend to use the terms interchangeably. If you have a hybrid car, your engine will have two types of motors (combustion motor and electric motor).