They (specifically neural networks trained on the cross-entropy error, with the logistic function as your activation function) work very well in classification

They (specifically neural networks trained on the cross-entropy error, with the logistic function as your activation function) work very well in classification tasks where you want to interpret the output as a probability, which is useful when you're more concerned with predicting across datasets rather than individual instances. My lab uses neural networks towards this end in predicting pharmacological activity across large libraries of molecules and all the prospective validation we've done indicates that neural networks are far and away the best-performing option out there for this domain as compared to e.g. SVMs or random forests.