As I understand it, an alternate reality doesn't need to correspond with the primary reality (the metaverse, the grid, the matrix, etc) whereas an augmented reality is essentially a HUD type setup for the primary reality.

As far as I can really tell, none. So I kind of use the terms interchangeably. Both are about layering virtual information over top of reality, as far as I know. I guess augmented is the superior term.

Augmented reality is an overlay onto what is truly there.

Alternate reality or Virtual Reality is a simulation of an entirely new universe.