So, I understand that normal maps add visual detail without actually modifying the geometry of an object, while displacement maps do change the model by adding real geometry.
But what exactly is a bump map? It was mentioned, but not really explained. Also, what are the typical use cases for each?
Displacement uses an image to move the vertices that exist. It doesn't add geometry.
Bumping uses images to creates the illusion of additional geometry. There are different methods of applying Bump. Two very common ones that are used in blender are height map, and Normal map. Height maps are Grey scale images and normal maps are color images.
Displacement Maps, Normal Maps, Height Maps, Bump Maps, all of them are like cousins or brothers to each other. They basically do similar stuff if not almost the same stuff, people use them interchangeably. You can use them for the same purpose which is to do "something" to the surface of a face. Displacement displaces if you set it to do the displacement, but beyond that, the other three or even the four of them tells the shader how light has to interreact with the surface.
In my mind it goes like this. A Normal Map is a Bump Map but fancier. The Normal Map uses that rainbow texture you see while the Bump Map uses just white, blacks and everything in between, so grays.
A Displacement Map will displace if you set it to do so, otherwise it's like a Bump Map.
Height Map is like a Displacement Map if you want it to push stuff up and down, but that's already what Displacement does. And if the Height and Displacement are not pushing anything, they are basically like a Bump Map.