The much obvious drawback of classical ML is that, you will have to engineer the features, and that requires good understanding of the data and domain as well. Further problem, is that as the number of features increase, model tuning can be challenging. For e.g., image classification using a random forest will be a sheer stupidity, considering how well CNN can handle the problem better.
However, Deep Learning or Neutral Network based models are not easily interpretable. For e.g. consider diagnosing a disease based upon physiological signals. The rules that a diagnostic expert (doctors) to will use are very straightforward. In that case, a classical ML model will play a better hand in crafting a rule based model. A black box model in medical industry is definitely not much appreciated.