Neural networks are impressive only in that they are able to give any kind of meaningful results at all. In the end, they are only a poor mimicry of real machine intelligence, and not much better, conceptually, than plain old nonlinear regression.

Nobody has been able to determine what the structure of a neural network should look like for any given problem (network type, number of nodes, layers, activation functions), how many iterations of the parameter optimization algorithm are needed to achieve "optimal" results, and how "learning" is actually stored in the network.

Statistical learning methods are obviously still useful, but I think the field is still wide open for something to emerge that is closer to true machine intelligence.