They deal with the non-linearities in the world in which we live. They handle noisy or missing data. They can work with large numbers of variables or parameters. They provide general solutions with good predictive accuracy. Neural networks are limited by the parameters assigned to the algorithms.
Inductive logic is beyond the purview of neural networks.
If artificial intelligence is to extend beyond the limits of human knowledge and neural networks are limited to human knowledge, then neural networks can be more efficient and accurate in applying human derived knowledge, however, will never go beyond the limits of human knowledge. I think mimicing the way we think is a fruitful road to AI, at least to begin with. I don't know all of the latest news and knowledge on this subject, but I think it is really the only feasible option we have at this point. I suspect that neural networks will be a very, very fruitful road to artificial intelligence. Being a potential Psych major, I know enough about psychology to have a heartfelt belief that all functions of the brain go back to the cells within our brain and how they interact.