It's one and the same, or rather, one is a special case of the other.

The homogeneous coordinate system used to represent affine transforms in R^n using linear transforms in R^(n+1) is exactly the same as what is used to represent projective transforms in the projective space P(R^n). This is famously exploited in 3D graphics where 4x4 matrices can represent linear and affine transforms and perspective projections (modulo the final w-division normalization step).

Affine transforms are a special case of projective transforms where the last row (or column depending on convention) vector is (0, ..., 0, 1).

Yes, I think I understand it (or am at least on the way to understanding it) now. Thanks!