Is there any domain where Bayesian Networks outperform neural networks?
One of the areas where Bayesian approaches are often used, is where one needs interpretability of the prediction system. You don't want to give doctors a Neural net and say that it's 95% accurate. You rather want to explain the assumptions your method makes, as well as the decision process the method uses. 

Similar area is when you have a strong prior domain knowledge and want to use it in the system.