How does employing evolutionary algorithms to design and train artificial neural networks have advantages over using the conventional backpropagation algorithms?
Unlike backpropagation, evolutionary algorithms do not require the objective function to be differential with respect to the parameters you aim to optimize. As a result, you can optimize "more things" in the network, such as activation functions or number of layers, which wouldn't be possible in the standard backpropagation.
Another advantage is that by defining the mutation and crossover functions, you can influence how the parameter search space should be explored.
Further to Franck's answer, there may be better optima (even global optima) that exist in the opposite direction to the gradient (which may be in the direction of some local optima). Evolutionary algorithms have scope to search the surrounding area, while backpropagation will always move in the direction of the gradient. With no guarantee (due to their randomness), evolutionary algorithms may be capable of finding solutions that backpropagation simply cannot.