This is more a remark then an answer.
One important application of projective/locally free resolution is the computation of intersection numbers. Let $X$ be a smooth projective variety over a field $k$, and $Z_1,Z_2$ subschemes of $X$ of complementary dimension. Let $L^*$ be a locally free resolution of $\mathcal{O}_{Z_1}$ then the intersection number $(Z_1 . Z_2)$ equals $\chi(X,L^* \otimes \mathcal{O}_{Z_2})$ (Serre's formula).
While the geometric meaning of $(Z_1 . Z_2)$ is clear (at least over the complex numbers): take a generic topological perturbation and count transversal intersections with orientation; the interpretation of $\chi(X,L^* \otimes \mathcal{O}_{Z_2})$ is less so.
We somehow trade the torsion sheaf $\mathcal{O}_{Z_1}$ for the more "global" object $L^*$ (with $Supp(L^i)=X$). We certainly need some global information about $X$ to compute the intersection, as the generic perturbations take place in $X$.
Note that if $Z_1 \sim Z_1'$ is an algebraic generic perturbation, such that $Z_1' \cap Z_2$ is transversal (and zero-dimensional), then $\chi(Z_1' \otimes Z_2)=h^0(Z_1' \otimes Z_2)=(Z_1 . Z_2)$. So there is a formal analogy between $L^*$ and a generic perturbation of $Z_1$.
Addendum: There is yet another link between deformations of $Z_1$ and resolutions of $\mathcal{O}_{Z_1}$: If one studies the local deformations of $Z_1$ inside $X$ one begins by deforming the surjection $L^{-1} \rightarrow \mathcal{O}_X$ to get a family of subschemes. To obtain a flat family one has to ensure that the relations do deform also, i.e. that there is a deformation of the presentation $L^{-2} \rightarrow L^{-1} \rightarrow \mathcal{O}_X$. I do not know, in this case we can always extend this deformation to the whole complex $L^*$. In any case, you can obtain deformations of $Z_1$ by deforming the maps $d_i$ keeping the relation $d^2 =0$ and the cohomology sheaves $H^{-i}(L^*), i>0$ fixed.