Aufzug is the more formal, technical term. For example, this wikipedia page contains references to technical and regulatory documents concerning elevators. None of them uses Fahrstuhl; in fact, the word does not appear on the page.
The two are semantically largely equivalent. This page claims that Fahrstuhl is more commonly used when referring to the actual cabin, but the example den Fahrstuhl kommen lassen would work with Aufzug as well.
Since it is not a technical term like Aufzug, Fahrstuhl is more colloquial.
Edit: Some commenters disagree in particular with the statement that Fahrstuhl is more colloquial. I don't have any reference for that. It was based on the use of Aufzug in formal contexts and my personal experience. It appears that others' experiences differ. Whether there is a correlation with region, social group or age is not clear yet (I'm a 54 years old college educated guy from the Hannover region). I'd welcome more comments with more information.