Well, with a human coming up with a PDE model (i.e. a system of equations) based on their intuition about what the causes of a given phenomenon are, they would be limited by their intuition/imagination about which variables are significant for the model. For all practical purposes, the variables that a modeler imagines to be significant are a subset of all the possible observable variables. A machine learning approach would learn the significant variable directly from the data, and is therefore not limited to imagination. There's an example of this mentioned in the article, in the context of ocean models:

“The algorithm picked up on additional terms,” Zanna said, producing a “beautiful” equation that “really represents some of the key properties of ocean currents, which are stretching, shearing and [rotating].”