Let $S$ be a scheme. Let $\mathcal{A}$ be a quasi-coherent graded $\mathcal{O}_S$-algebra. Assume that $\mathcal{O}_ S = \mathcal{A}_0$. The affine cone associated to $\mathcal{O}_S$-algebra $\mathcal{A}$ is the $S$-scheme $C:= \underline{\mathop{\mathrm{Spec}}}_ S(\mathcal{A})$. I understand the definition and my geometric intuition is that it generalizes the geometric construction of a "ray" space of projective space; nevertheless I'm far away from fully comprehending what are the advantages and usage of the cone construction in algebraic geometry.
Which kind of problems can be conceptually attacked effectively using this construction and what is the intuition making it a useful tool one should have? Is it more than an introducing model example of Grothendieck's relative point of view?
Can its role in algebraic geometry be compared with the role of cone construction in topology? In that setting a cone provides a space, which contains the original space, but has a bunch of nice topological properties, which making it interesting for homotopy theory.
In algebraic geometry we have also a zero section $i_0: S \to C$. The point is: what is the bunch of properties making the cone in a certain way easier to work with (possibly like better control over regularity, certain nice properties for birational geometry, from intersection theoretical viewpoint, and so on, I don't know; to gather at least the most important ones is precisely the motivation for posting this question), while on the other hand keeping enough information about the original space that we can draw back conclusions about it (and so that justify its advantages)?