When does decision tree perform better than the neural network?
Neural Networks, in my experience have several hyper-parameters (number of layers, neurons per layer, activation functions, optimizers, regularizers, etc.) and are very hard in finding the best configuration for each task. In fact in most cases it's not even worth it trying to find the optimal configuration as other classifiers can outperform Neural Networks with default hyper-parameters. Furthermore, NNs require caution as they are prone to overfitting.

For most tasks where you deal with structured data, I've found tree-based algorithms (especially boosted ones) to outperform NNs. 

Some NN architectures are state-of-the-art tasks where we have a lot of unstructured data (e.g. CNNs for image-related tasks).

Finally, I'd like to say that there are no absolutes (e.g. SVMs will alawys outperform DTs). There is also a theorem along these lines: No Free Lunch Theorem.