Here in the USA most people seem to use these two terms interchangeably, but there are those who argue that "motor" should pertain to electric motors, whereas "engine" should pertain to non-electric motors. What distinctions exist in the rest of the world, and why?

They are not interchangeable in the UK. A motor is a device that provides motive power and can be of any kind.

An engine, while often used to refer to a motor, can also mean other kinds of device than those providing motive power.

Other common uses are that of something used to achieve a purpose, and that of a wheeled vehicle consisting of a self-propelled engine that is used to draw trains along railway tracks or even roads.

It can also, incidentally, mean the face of a watch.

What is the face of a watch? Engine?

In fact I was not quite accurate in my commentary. The full description would be "engine-turned" or maybe "rose-engine turned" - to describe the kind of engine used.

A good example of a classic watch with a rose-engine turned dial is here.

When talking about railways/railroads, there are three main types of motive power; steam (now alas almost defunct), diesel, and electric. For all of these the locomotive is known as the "engine". If we were to talk about the "motor", we'd be referring to the actual diesel- or electric-powered gubbins that drives the wheels, rather than the whole locomotive.

Rockets are also motors, and computer programs can be engines (e.g. inference engines). Charles Babbage's mechanical computer was called a difference engine.

quote:

Charles Babbage's mechanical computer was called a difference engine.

Sadly the device never worked properly (Babbage was part of the problem since he was always looking for the next development). His difference engine (basically a calculator) he abandoned and started work on his analytical engine - which was a programmable computer using Jaquard punch cards. This was over a hundred years before the world's first programmable computer was invented - at Bletchley Park during WW2.

Although Babbage never managed even to get a prototype analytical engine operating, he did get a prototype of the difference engine running and the remains of it can still be seen in London's Science Museum. Engineers at the museum reconstructed the difference engine using Babbage's original plans and, when they tried it out in the early 1990s, it worked perfectly and returned results to 31 digits of accuracy - a very high level.

You can see pictures of both the original Babbage machine and the reproduction machine here: http://www.sciencemuseum.org.uk/on-line/treasure/objects/1862-89.asp