The functor of points approach has the advantage of making certain constructions much simpler (e.g. the fibered product in the category of schemes), and eliminating the need for certain constructions like the Zariski spectrum. In his famous 1973 Buffalo Colloquium talk, Alexander Grothendieck urged that his earlier definition of scheme via locally ringed spaces should be abandoned in favour of the functorial point of view.

Of course, the above discussion generalizes to other types of geometry and even higher geometry.

Example

The functor from commutative rings to sets which sends a ring, $R$, to the set of simultaneous solutions in $R^n$ of a set of polynomials, $f_1, \ldots, f_k$ in $\mathbb{Z}[t_1, \ldots,t_n]$ corresponds to the affine scheme $X = Spec(\mathbb{Z}[t_1, \ldots,t_n]/(f_1, \ldots,f_k))$. These $R$-points are then equivalently the hom-space

$Hom_{schemes}(Spec(R), X).$

The functor which sends $R$ to the points of the projective space $\mathbb{P}^n_R$ corresponds to a non-affine scheme.

Alexander Grothendieck, Introduction to functorial algebraic geometry, part 1: affine algebraic geometry, summer school in Buffalo, 1973, lecture notes by Federico Gaeta, pdf scan.