The two constructions are not quite equivalent. Let me write $\mathbf BG$ for the stack and $B_\bullet G$ for the simplicial scheme to better distinguish between them. There is a third relevant player, $BG$, which is the presheaf of ∞-groupoids on $C$ presented by $B_\bullet G$.

The precise relation between these three objects is the following:

- $\mathbf BG$ is the fppf sheafification of $BG$
- $BG$ is the colimit of the simplicial object $B_\bullet G$
- $B_\bullet G$ is the Čech nerve (= $0$-coskeleton) of either $\mathrm{Spec}(k) \to \mathbf BG$ or $\mathrm{Spec}(k) \to BG$

In more detail, algebraic stacks over $k$ (in the sense of Artin, say) form a full subcategory of the $2$-category of fppf sheaves of groupoids (classically, "stacks in groupoids") on $C$, which is itself a full subcategory of the ∞-category of presheaves of ∞-groupoids on $C$. Thus, both $\mathbf BG$ and $BG$ live in this ∞-category (in fact they both belong to the subcategory of presheaves of groupoids), and one is the fppf sheafification of the other. The étale sheafification suffices if $G$ is smooth.

The reason $\mathbf BG$ and $BG$ are not the same is that there is a unique homotopy class of map $X\to BG$ from any scheme $X$, but homotopy classes of maps $X\to \mathbf BG$ are in bijection with isomorphism classes of $G$-torsors on $X$. In fact $BG$ is the full subpresheaf of $\mathbf BG$ spanned by the trivial $G$-torsors.

(Added details about 2.)
$BG$ being the colimit of $B_\bullet G$ is the manner in which simplicial sets give rise to ∞-groupoids. Of course, one must define ∞-groupoids at some point and one way to do that is to start with simplicial sets and invert weak equivalences, so that we have a localization functor {simplicial sets} → {∞-groupoids}. Once higher category theory is set up however, it turns out that this functor is the composition of the inclusion of simplicial sets into simplicial ∞-groupoids and of the colimit over $\Delta^{op}$ functor (this is the standard fact that a simplicial set is canonically the homotopy colimit of itself). In practice this is often a more useful way to think about it.