Lets say i need to check if two AABBs are colliding with each other. I would simply compare their corners. When am i gonna need SAT(heavy vector calculations) to do the same thing?
SAT is for convex polygons, AABB is for axis aligned bounding boxes (obviously). If you only have the latter ones, then you don't need to use SAT, but if you need something that's more precise, then SAT is the way to go (polygons can enclose objects more precisely, though it requires you to make them manually or use some sort of algorithm to generate them)
Usually AABB is used for the broad phase, and SAT is for the narrow phase.