In everyday language I would say Palewriter is correct: "engine" is generally only used for combustion engines, ie. ones which burn a fuel. The word combinations "petrol engine", "diesel engine" and "steam engine" trip very readily off the tongue. I think that in the past, it was used more in the sense that we would use "machine" today, for example a siege engine. I think talking about search engines would only complicate things!
"Motor" is something which produces movement. This could include engines, as in "motor car", but normally these days the assumption is that it is an electric motor, unless otherwise stated or implied. I have the feeling that "motor" was used much more often for what I would call an engine in the first half of last century, but since I wasn’t around at the time I can’t be sure without doing a lot of research.
Edited to add: Just look at that - no replies for eight hours, then two within a minute of each other. Still, I’m glad I haven’t entirely contradicted the famous Mr. Quinlon.
JtW.