Traditional ML methods exist for all of the things we use neural nets for, but none of them are as effective, for a plethora of reasons, but one of the biggest reasons is how much training data they can handle. If you have to invert an NxN matrix, for example, where N is the size of your training set, you aren't getting very far. But a neural net scales to datasets containing billions of samples, and can be adapted to multiple domains that previously had their own special techniques. The bottleneck was being able to train them, and letting go of restrictions like provable optimality. Once we could train them, we quickly discovered that scaling to larger datasets produced models that dominated everything else.