I should have been more careful. I was meaning to say the main differences in *how calculus and analysis are taught*; I was not meaning to make any claim about what the two meant historically. In this sense, analysis (real, complex, Fourier, and functional) is most certainly about proving theorems. At the undergraduate level, numerical analysis doesn’t seem to be a proper subcamp of analysis, though this could simply be because of how it is taught at my university where only calculus, linear algebra and ODEs are prerequisite courses.
To stop with Newton and Leibniz is an injustice, however. Cauchy and Weierstrass and others of the 19th century are equally worth mention in the development of modern analysis. In my opinion, the 1800's is when analysis (using this as a pure mathematician would) got underway.
My initial response was simply to indicate that “advanced calculus”, as a university level course, typically functions as a bridge to learning differential geometry and treating vector calculus techniques more generally.