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).
The shading model computes the reflected light at a single point, by accounting for the ray and surface directions and reflectance characteristics.

The interpolation method is made to compute the reflected light between points where the normal is known. You distinguish

* Gouraud shading: you evaluate the light color using the reflectance model at the known points, and interpolate on the color;
* Phong shading: you interpolate the normals and apply the reflectance model at every interpolated point.

Phong is more costly but more realistic.