1 Monty: Since our train doesn't leave until the evening, we have time to take a taxicab out to the Pyramids.
2 Prof. Jones: Ooh! Did you know there's a secret chamber inside the Great Sphinx?
3 Monty: That's an unconfirmed speculation.
4 Prof. Jones: Your mother and I found it one time. It's pretty private in there, let me tell you!

According to Wikipedia, taxicab is portmanteau contraction of the words taximeter (a mechanical device used to record fares based on time and/or distance) and cabriolet (a type of light horse-drawn carriage).

I don't know about other countries, but here in Australia the shortened words taxi and cab are used about equally to refer to the modern vehicles, and are readily interchanged in conversation.