@Matt. I'm not really an expert here, and maybe I misunderstood the question, so here's my low tech explanation, and someone can add details or correct me. First I think the question really was "infinitesimal" v. "local" deformations, as I'm not sure what a "global" deformation is.  I interpret "infinitesimal" deformations as elements of a zariski tangent space.  If $f:{\bf C}^a\to {\bf C}^b$ is a map, then the tangent space at $v\in V=f^{-1}(0)$ is $\ker df$.  At a smooth (i.e.submersion) point,  $V$ is locally isomorphic to $\ker df$, and so every infinitesimal deformation is locally deformable. But at a singular point some formal tangent vectors (i.e. vectors in the zariski tangent space)  aren't tangent vectors, e.g. in the example of $z^2$. 

Typically one is trying to deform some structure, i.e. move around in some kind of  moduli space, and there is a group acting, so that the zariski tangent space to the moduli space at a point is a quotient $\ker df_v/im~ g$, where $g$ is the orbit map. Frequently this quotient is identified with $H^1$ of some complex ($g= d_0, df_v= d_1$), so that the infinitesimal deformations near $v$ are identified with some $H^1$. Kuranishi's method (for example) is a general trick which locally describes the moduli space near $v$  as cut out by a non-linear map $k:H^1\to H^2$, so that   $k^{-1}(0)$ describes the local deformations inside the infinitesimal deformations $H^1$.  In many cases the quadratic part of $k$  is given algebraically (i.e. by cup products) and in really nice cases this is enough to completely describe the local deformations (hence Felipe's comment).

All this fits in a larger context (e.g. Artin' paper  http://www.ams.org/mathscinet-getitem?mr=232018   and $R[t]/t^n$, etc.)