Question: My class notes have made it clear that these two are not the same. I understand that the Phong Illumination Model is ambiant+diffuse+specular, but where does Phong Shading come into play when it comes to lighting? What about BRDFs?

Answer: 
An illumination model (or reflectance model) tells you for a given surface point, light, and viewer, the amount of light that is reflected towards the viewer. The Phong illumination model does this with the three light components you described and is a special case of a BRDF (only the diffuse and specular part; the ambient part cannot be modeled with a standard BRDF). The illumination model is completely continuous and only defined in physical space (although they need not adhere to a physical illumination model).

A shading model is a strategy to produce an image based on the shape and and illumination model. More specifically, it defines where to evaluate the illumination model and what to do with the result. The Phong shading model evaluates the illumination model at every pixel and uses the result directly as the pixel color. Other shading models evaluate the illumination model only at vertices and blend the colors (Gouraud) or once per face (flat).
Phong Shading is a method used to calculate the shading on an object.
To calculate the shading of a point on an object, the *normals of the vertices* are interpolated.
In comparison, flat shading only uses the normal of the polygon. Phong shading will result in smoother, more curved edges while at the same time not requiring a lot of polygons.

Also see [this article](https://opengl-notes.readthedocs.io/en/latest/topics/lighting/shading.html) and especially the picture.