I have heard many people debate this topic. Personally I believed that a motor was electrical and an engine was, at the most rudimentary, a device that converts thermal energy to mechanical energy (steam engine, internal combustion engine). I decided to do some research myself to see which is correct. Please note that I was wrong in my previous definition.
From Wikipedia, the disambiguous definition for Engine:
"An engine is a device that converts potential energy into mechanical work." It may also refer to: heat engine, internal combustion engine, external combustion engine, Carnot heat engine, Stirling engine
The Wikipedia definition for Motor:
A Motor is a device that creates motion. It usually refers to an engine of some kind. It may also specifically refer to: AC motor, DC motor, Electrostatic motor, Engines, Servo motor, Starter motor, Stepper motor, Actuator, Hydraulic motor, Rocket motor, Molecular motor, Nanomotor, Pneumatic motor
From Dictionary.com:
motor
mid-15c., from L. motor, lit. "mover," from movere "to move" (see move). From 15c. as "controller, prime mover" (in reference to God); sense of "agent or force that produces mechanical motion" is first recorded 1660s; that of "machine that supplies motive power" is from 1856.
With explosive use 20c. as a comb. form of motor-car (1895). Motor-boat is from 1902. First record of slang motor-mouth "fast-talking person" is from 1971.
So from what I've read, I believe a motor is any device that causes motion -- in the case of vehicles, this could be either an electrical motor or an internal combustion engine. Regardless of what's 'under the hood', that power source is called a motor, since it is causing motion of the vehicle. I guess I need to learn to accept that it is correct to call that thing under the hood a 'motor' and stop 'correcting' people!
On an interesting side note, I also tried to find the first internal combustion engine, which appears to be one powered by gunpowder
17th century: Christiaan Huygens designs gunpowder to drive water pumps, to supply 3000 cubic meters of water/day for the Versailles palace gardens, essentially creating the first idea of a rudimentary internal combustion piston engine.
Let the debate begin!
Get out of the HOUSE and drive your AA!! 
