I'm a little unclear on what Affine Maps are. Are they just maps that look linear but actually are not because they don't preserve sums (don't go through the origin)? Or are they more important?

I think affine maps are just a broader categorization of maps. All linear maps are affine, but not all affine maps are linear.

Yeah, "like a linear map, but not preserving the origin" is a good way to think of an affine map.