There is really no mathematical advantage that NNs have over other approaches namely statistical, optimization and formal symbolic approaches/algorithms which are (imho) better engineering tools as they are more amenable to analysis. At the same time NNs are not inferior to other approaches, mathematically. The only advantage of NNs are sounding cool and being superficially similar to our brains. Consider this simple and often repeated analogy, sticking a beak and feathers on your toy airplane won't make it go faster. In real AI research, NNs are just a bullet point in the huge field of Machine Learning (http://ijcai-11.iiia.csic.es/calls/call_for_papers).
AI people should stop throwing around cool names and instead build things which are real (Please do not start another AI winter.) Watson is a refreshing step in the right direction.
In this case, neural networks have the advantage of looking similar enough to networks of memristors that you may be able to cram a big neural net into a small, low-power chip. It's not a huge breakthrough in AI, but prove very handy for some applications.
AI people should stop throwing around cool names and instead build things which are real (Please do not start another AI winter.) Watson is a refreshing step in the right direction.