In what way do genetic algorithms "not require training"? The evolution of the algo is the training, no?
GAs can be quite useful in situations where there are tons of very bad local minima/maxima and you desire the global minima/maxima. Unfortunately, neural networks have lots of really great local minima/maxima and the state space is so large that you'll likely never get the global minima/maxima (and you wouldn't know it if you did). This is why "Neuroevolution" of neural network weights hasn't really caught on.
One of the beautiful aspects of Evolutionary Computation is how you can use it as a generative mechanism. A fairly simple fitness function can produce complex and intriguing outputs.