It depends on the class, and almost certainly on the department.

But actual history classes in the classical studies department at my school were taught by history professors (meaning professors from the actual history department) who had an adjunct appointment to classical studies. The modern language departments did the same thing - history courses would be taught by a specialist in German or French (or whatever) history, not from someone within the language department itself.

And this procedure wasn’t limited to history courses - Greek sculpture would be taught by someone from the fine arts department; religion courses would be taught by someone religious studies; philosophy from someone in the philosophy department; etc.

I'm not sure if that is exactly what you are looking for.

Ancient historian here. Depends on the school. Sometimes ancient historians abide in history departments, sometimes in classics. Usually if there are actually trained historians available (regardless of their department of residence), they'll teach the history courses; if not, the classicists will take over.

The main difference I've observed between the two is you'll generally get a more historically focused, better sourced, and more well-rounded course from a historian. Classicists, depending on their background and expertise, tend to work literature into their courses whenever possible (surprise!), and they'll usually only teach to the material they were taught in graduate school, which means some stuff—particularly where archaeology is involved—will only date to whenever they graduated.

I was taught Greek and Roman history by classicists in my undergrad, and I was shocked when I got to graduate school by how out of date some of their info was—it was very obvious they finished in the late eighties! They also didn't teach me anything about bronze or Iron Age Greece, or about Late Antiquity, because classicists' training often (not always) begins with Homer and stops at Tacitus.

As such, a Roman history class from a classicist will probably find a way to make you read the Aeneid and will start to peter out pretty quickly once you get to Commodus. If the classicist is a historiographer, though, they'll probably have you read some weird and obscure shit. Bonus points if they spend more than two days on the 3rd century AD.