The sole difference between these two transformations is in the last line of the transformation matrix.
The projective transformation does not preserve parallelism, length, and angle. But it still preserves collinearity and incidence.
Since the affine transformation is a special case of the projective transformation (the first two elements of the last line should be zeros), it has the same properties. However unlike projective transformation, it preserves parallelism.
Projective transformation can be represented as transformation of an arbitrary quadrangle (i.e. system of four points) into another one. Affine transformation is a transformation of a triangle. Since the last row of a matrix is zeroed, three points are enough.