Here is a simple example. Someone asked me if the unit disc is an affine variety. the answer is no. To see why not, assume yes, and take the projective closure. then one gets a projective curve which is a compact 2 dimensional surface with some points identified, and which differs from the original surface by at most adding a finite number of points. But this is impossible. No compact surface can be reduced to a disc by removing a finite number of points, even topologically, except for removing one point from P^1. But that does not give the disc by Liouville's theorem.
A more significant and pervasive example is the fact that at every singular point of an affine variety, the tangent cone determines a projective variety. Thus projective geometry is the local aspect of affine geometry. Put another way, blowing up an affine variety, at a point say, introduces projective geometry into it as a picture of its infinitesimal structure.
One can sometimes use this trick to compute the degree of a proper map of affine or other varieties, by restricting to the behavior over the projective normal bundle of a single fiber. See e.g. Friedman - Smith (Inventiones, 67, (1982)), who compute the degree of the prym map by showing that a single projective fiber of the proper prym map is embedded in projective space by the derivative of the prym map acting on the normal bundle to the fiber.