Automatic differentation is "symbolic" in the sense that the resulting derivatives are exact, in the same way that you get exact derivatives if you differentiate by hand and implement the resulting expression. There is no approximation.

Numeric differentiation has a specific meaning - which is computing derivatives via finite differencing.

There's three ways to compute derivatives; for whatever reason most people only know about symbolic (take the derivative by hand, implement it in C++) and numeric (implement your cost function, do finite differences on each parameter to determine the gradient) differentiation. Automatic differentiation is a totally different way to take derivatives. The Wikipedia article about it is fairly good.

We didn't pick the name.