He might be using "exponential" to mean "superlinear", which seems to be the sense a lot of my students try to use it in (as well as non-technical people).
Now that you mention it, I guess non-technical people do tend to use that definition. I had never thought about it before. It's unfortunate, considering there's a big difference between, say, quadratic and exponential growth. In fact, it tends to be a more impactful difference than between linear and quadratic, especially regarding algorithms.
I am not aware of the difference. Is it that exponential means something like c^n, superlinear is n^c, and linear cn?
Superlinear is anything above linear. This includes pseudolinear (e.g., O(nlog n)). Exponential is O(m^n). Linear is O(n).
Edit: And polynomial is O(n^m).
What Locke1689 said. If you're not familiar with big O notation, linear is like f(x) = 2x, quadratic is like f(x) = x^2, and exponential is like f(x) = 2^x. There's a huge difference between quadratic and exponential: exponential grows significantly faster as x increases. For computing, the difference is significant. Cobham's thesis says that polynomial algorithms (which includes quadratic) are reasonable to perform, while exponential algorithms aren't.