Evolutionary algorithms and neural networks can both be used for similar domains for powerful computations because evolutionary algorithms are meta machine learning algorithms used for optimization and neural networks are used for model training.
In machine learning problems, you typically have two components:
a) The model (function class, etc)
b) Methods of fitting the model (optimization algorithms)
The neural network is a model in which we provide 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 i.e. fitting the model. Again, there are some canonical models that go with evolutionary algorithms: for example, evolutionary programming typically tries to optimize overall programs of a particular type. However, they are responsible for finding a way to 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 to get a set of parameters.