I guess, it is all dependent on which side of the fence you are on.
An ancient historian may argue that there is no difference and that classicist simply choose to be termed "Classicist" out of tradition. A classicist may argue that their education is more traditional and beyond the mere memorization of facts.
While classics and history majors may attend the same lectures, classes, conferences, and so on; there is something to be said for a "Classics" education. Many view that Classics as being a gateway into other areas of study such as politic science, law, computers, education, and so on. It has been more heavily believe that a Classics education is more well rounded and suitable to an individual that may or may no know their end result- but knows "Dr." or "Supreme Court Justice" is in their future.
However, Classics students may have a sense of elitism that stems from their peers or even professors or advisers. However, it should be considered that most of the populace does not even care that there is a difference nor would they hire one over the other.
As contemporary student within academia, I have been asked several times what the difference is between Classics and Ancient History. I would have to respond that it is the language. I feel that Ancient History majors are not required nor do they have to train extensively in the ancient languages or texts.
As a Classicist one is expected to take the time to translate it themselves and then begin their proper work on philosophy, war tactics, and so on- using the translation they rendered. This is not the case for Ancient Historian who may pick up a well reputed and respected translation and read it in half or a quarter of the time and then begin their work.
I would also argue that Classicist are more concerned with the everyday life and society of ancient civilizations- almost closer to an anthropologist. And, that Classicist have a keener interest in philology. While, a historian may or may not concern themselves with these aspects.
More may be read on the subject at:Examiner: Classicist versus Historian