Ever since I first read about that it seems to me that autodiff really is symbolic differentiation implemented in an efficient way for computer programs.

From my point of view if you are not approximating the derivative by a small, non zero "h" it is, from a mathematical perspective, symbolic differentiation.

However, in the practical world there is a big difference in performance between working with symbolic expressions and their derivatives in a computer (for instance in Mathematica or SymPy) or using autodiff.

That is why my opinion today is that from a "mathematical" point of view there is no difference, from a "computer science" point of view there is a difference, which I summarize as "autodiff is an efficient implementation of symbolic derivatives".