Let $A$ be a local ring with the unique maximal ideal $\mathfrak{m}$. The punctured spectrum of $A$ is the open subset $\text{Spec}(A)\setminus \{\mathfrak{m}\}$. I have seen many papers (for instance Horrocks' papers) studying vector bundles over algebraic varieties (in particular, projective spaces) by putting them over a punctured spectrum of a local ring.

However, I am wondering why punctured spectrum is better than varieties in this satiation. I feel that geometric pictures of varieties are clearer than that of a punctured spectrum. More essential, are the categories of coherent sheaves over a variety and its punctured spectrum equivalent? How much information about coherent sheave (in particular, vector bundles) can be recovered from punctured spectrums? For second question, I am thinking examples that reflect some relations between those two gadgets.