Put another way: with evolution you have to stumble around blindly in parameter space and rely on selection to keep you moving in the right direction. With the gradient descent that neural networks use, you get, essentially for free, knowledge of the (locally) best direction to move in parameter space.
The bigger the models, the more this matters. Modern neural networks have millions or even billions of parameters, and that's been crucial to their expressive power. Good luck learning a program tree with a billion nodes using evolution. It might take 4.54 billion years.