I'm an absolute beginner in algebraic geometry so please excuse if the following question is somewhat superficial.
Hartshorne's "Algebraic geometry" begins with the definition of (quasi-)affine and (quasi-)projective varieties over some fixed algebraically closed field. At a first glance, these seem to be quite different, so that I would have expected that one would pose questions either on quasi-affine or on quasi-projective varieties.
However, Hartshorne then defines a variety to be either a quasi-affine or a quasi-projective variety. These varieties (together with certain continuous and in some sense regular maps) then form the category of varieties.
Here is my question: Is the above definition natural in the sense that we really want to compare quasi-affine and quasi-projective varieties or at least study them both at the same time?
For instance, is there a (non-trivial) example of a quasi-affine variety which is isomorphic in the above category to a quasi-projective variety? If not, isn't this "unifying" definition a bit artificial?