Neural networks are known to be generally better modeling techniques as compared to tree-based models (such as decision trees). Are there any exceptions to this?
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Hard to say in general. Speaking from my own experience and by looking at which models win Kaggle competitions (see here and here), I would say tree-based models e.g. Random Forests, Decision Trees, Gradient Boosting are favorable over neural networks when working with low-dimensional data and easy interpretable features (usually simple tabular data with numeric, ordinal or categorical features).
Whereas when working with everything high-dimensional like images, text, time series or other data with non-trivial features, I would recommend neural networks.
Of course there might be exceptions and the future may prove me wrong.
