I have heard the terms "Newtonian mechanics" and "classical mechanics" used interchangeably, but is there a difference between them? If so, what is it?
In advanced physics circles, "classical mechanics" always means something very specific, and it doesn't mean "Newtonian." Classical means not quantum. Newton's theory of gravity is just one example of a classical theory. Einstein's general relativity is also a classical theory. Any theory which says gravitation is due to the exchange of gravitons would be a quantum theory. In this usage of the term, all theories are either classical or quantum. The quantum theories are the more fundamental ones, and the classical theories are in some sense approximations of them.
The term "Classical mechanics" sometimes describes brunch of physics or a course, which focuses on Lagrange and Hamiltonian formalism. This are a little bit advanced tools, which are widely used (Euler-Lagrange equations etc...).
Because this (mathematical) tools are a little bit harder, they are often taught on the example of something, which we understand very good and it's quite simple and this is simple mechanic (pendulums, objects on slopes, spinning-tops...). This course at uni is often titled "Classical mechanics". It focuses on describing Newtonian mechanics, but with different tools, which can be than applied on other fields of physics (often quantuum etc). Notable classical mechanics problems are spinning-top (precession, nutation), all components of Couriolis force, shape of pendulum, central potenitals, Euler-Lagrange equations, Hamilton's formalism, double pendulum...