Yes, but in practice the difference is vast. You can apply automatic differentiation to (almost) arbitrary code with conditional branches, arbitrary inputs (by

Yes, but in practice the difference is vast. You can apply automatic differentiation to (almost) arbitrary code with conditional branches, arbitrary inputs (by treating them as unknown constants wrt to the independent variables), recursion and for/while loops with an unknown number of loops, etc. Trying to describe all that using piecewise or recursive functions and then symbolically differentiate it is either a nightmare or impossible. Hence, automatic differentiation.