[Michael Hill]

The best explanation I've seen basically goes as follows:

All engines are motors, but not all motors are engines.

To be an engine, it must be a machine/mechanism (i.e., moving parts) that serves as an energy source.

Examples:

Electric Motor - It has moving parts, but the energy source (the battery) is not part of it, thus it isn't an engine.

Combustion Engine - It has moving parts and gasoline is burned inside, therefore it IS an engine.

Solid Rocket Motor - It serves as an energy source (burning fuel inside of itself), but it has no moving parts, thus it isn't an engine.

Liquid Rocket Engine - It serves as an energy source (burning fuel inside of itself), and there are tons of moving parts (turbomachinery), therefore it IS an engine.