What is an interesting example of that? Things like $Spec(K) \to Spec(L)$ do not count cause they are not interesting.
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Dear Dung, a pleasantly geometric example of a quasifinite, separated, but not finite morphism is the projection of the hyperbola $xy=1$ in the affine $x,y$ plane on the $x$axis. Its image is the affine line minus the origin. It is clearly quasifinite (even injective) but not finite, since its image is not closed . (Also, morphisms between affine schemes are separated) 

