Are evolutionary algorithms and neural networks used in the same domains?

Here is the deal: in machine learning problems, you typically have two components:

a) The model (function class, etc)

b) Methods of fitting the model (optimizaiton algorithms)

Neural networks are a model: given a layout and a setting of weights, the neural net produces some output. There exist some canonical methods of fitting neural nets, such as backpropagation, contrastive divergence, etc. However, the big point of neural networks is that if someone gave you the 'right' weights, you'd do well on the problem.

Evolutionary algorithms address the second part -- fitting the model. Again, there are some canonical models that go with evolutionary algorithms: for example, evolutionary programming typically tries to optimize over all programs of a particular type. However, EAs are essentially a way of finding the right parameter values for a particular model. Usually, you write your model parameters in such a way that the crossover operation is a reasonable thing to do and turn the EA crank to get a reasonable setting of parameters out.

Now, you could, for example, use evolutionary algorithms to train a neural network and I'm sure it's been done. However, the critical bit that EA require to work is that the crossover operation must be a reasonable thing to do -- by taking part of the parameters from one reasonable setting and the rest from another reasonable setting, you'll often end up with an even better parameter setting. Most times EA is used, this is not the case and it ends up being something like simulated annealing, only more confusing and inefficient.

Problems that require "intuition" are better suited to ANNs, for example hand writing recognition. You train a neural network with a huge amount of input and rate it until you're done (this takes a long time), but afterwards you have a blackbox algorithm/system that can "guess" the hand writing, so you keep your little brain and use it as a module for many years or something. Because training a quality ANN for a complex problem can take months I'm worst case, and luck.

Most other evolutionary algorithms "calculate" an adhoc solution on the spot, in a sort of hill climbing pattern.

Also as pointed out in another answer, during runtime an ANN can "guess" faster than most other evolutionary algorithms can "calculate". However one must be careful, since the ANN is just "guessing" an it might be wrong.