What is the difference between advanced calculus and real analysis?

The content of any course depends on where you take it--- even two courses with the title "real analysis" at different schools can cover different material (or the same material, but at different levels of depth).

But yeah, generally speaking, "real analysis" and "advanced calculus" are synonyms. Schools never offer courses with *both* names, and whichever one they do offer, it is probably a class that covers the subject matter of calculus, but in a way that emphasizes the logical structure of the material (in particular, precise definitions and proofs) over just doing calculation.

My impression is that "advanced calculus" is an "older" name for this topic, and that "real analysis" is a somewhat "newer" name for the same topic. At least, most textbooks currently written in this area seem to have titles with "real analysis" in them, and titles including the phrase "advanced calculus" are less common. (There are a number of popular books with "advanced calculus" in the title, but all of the ones I've seen or used are reprints/updates of books originally written decades ago.)

There have been similar shifts in other course names. What is mostly called "complex analysis" now in course titles and textbooks, used to be called "function theory" (sometimes "analytic function theory" or "complex function theory"), or "complex variables". You still see some courses and textbooks with "variables" in the title, but like "advanced calculus", it seems to be on the way out, and not on the way in. The trend seems to be toward "complex analysis."  hope it helps