Let $Vect_{n}(C)$ the moduli stack of vector bundles $V$ of rank $n$ over a smooth curve $C$ of genus $g$. It is well known that $Vect_{n}(C)$ is a smooth stack of dimension $n^{2}(g-1)=\dim(H^{0}(C,End(V)))-\dim(H^{1}(C,End(V)))$
Intuitively, the first term counts the space of automorphisms and the second the space of deformations. However my question is basically why is necesary to consider deformations when building moduli stacks. My guess is that $V$ and its deformation $V_{\varepsilon}$ are going to have different topological data (chern classes for example?).
Otherwise, I don't see why you have to consider them as different points in the moduli stack as the difference between them would be somewhat formal and not real (with formal I refer, for example, to the information of the formal neighbourhood of schemes which is not captured at level of varieties (and should not be captured at the level of this example as it deals with algebraic curves as varieties, not as schemes)).