Regression, both linear and logistic are from the mid 1800s to early 1900s. Neural networks, at least the basics are from around 1950.
What has really changed is the engineering, the data volume and the number of fields we can apply the mathematics to. The math itself (or what is the basis of AI) is really old.
and it was only in the last decade that the vanishing gradients problem was tamed.
my impression is that ML researchers were stumbling along in the mathematical dark, until they hit a combination (deep neural nets trained via stochastic gradient descent with ReLU activation) that worked like magic and ended the AI winter.
One of the big pieces was Schmidhuber's lab's highway nets, done ~30 years ago, but just didn't land until a more limited version was rediscovered.