One of the key differences (amongst many others) is the manual effort involved in feature extraction.

In your example, logistic regression (or other ML techniques) can be used to identify variable importance, but the inference, interpretation and engineering must be done manually. End objective is the association of (right) independent variables to the dependent one, after eliminating all extraneous ones.

With DL, this manual effort to establish the connection can be greatly reduced, as shown by the facial recognition example in the article. Hope this clarifies.