These notions are equivalent: to be precise, given any sheaf of abelian groups $M$ on $X$, there is a canonical bijection between $\mathbb{C}_X$-module structures on $M$ and sheaf of $\mathbb{C}$-vector space structures on $M$ and this bijection preserves morphisms of sheaves and thus gives an isomorphism between the two categories.

The idea behind this is simple. First, any sheaf of $\mathbb{C}_X$-modules is a sheaf of $\mathbb{C}$-vector spaces since $\mathbb{C}_X(U)$ is a $\mathbb{C}$-algebra (and the restriction maps of $\mathbb{C}_X$ are $\mathbb{C}$-algebra homomorphisms). But inversely, if you have a sheaf $M$ of $\mathbb{C}$-vector spaces, this gives you a way to multiply sections by locally constant functions. How? Well, suppose $x\in M(U)$ and $f$ is a locally constant function $U\to\mathbb{C}$. Cover $U$ by open sets $V_i$ such that $f$ is constant on each $V_i$, say with value $c_i$. To multiply $x$ by $f$, you can now take the restriction of $x$ to each $M(V_i)$, multiply these by the scalars $c_i$, and then glue them back together to get a section in $M(U)$. (The sections $c_i x|_{V_i}\in M(V_i)$ are compatible on the intersections $V_i\cap V_j$ because $c_i$ must be equal to $c_j$ if $V_i\cap V_j$ is nonempty.)

Of course there are a lot of details to be checked to verify that this really gives an isomorphism of categories, but they are straightforward. The point is that for a sheaf, being able to multiply by scalars from $\mathbb{C}$ is equivalent to being able to multiply by locally constant functions since using the sheaf condition you can *locally* multiply by the constant values and then glue the results together.

(If you have a bit of machinery set up, here is one nice way to verify the details. A $\mathbb{C}_X$-module structure on $M$ can be described as a homomorphism of sheaves of rings $\mathbb{C}_X\to \mathcal{Hom}(M,M)$. Also a $\mathbb{C}$-vector space structure can be described as a homomorphism of *presheaves* of rings $\mathbb{C}_X^0\to\mathcal{Hom}(M,M)$, where $\mathbb{C}_X^0$ is the constant presheaf with value $\mathbb{C}$ (i.e., $\mathbb{C}_X^0(U)=\mathbb{C}$ for all $U$ and the restriction maps are the identity). But now since $\mathcal{Hom}(M,M)$ is a sheaf and $\mathbb{C}_X$ is the sheafification of $\mathbb{C}_X^0$, every presheaf morphism $\mathbb{C}_X^0\to\mathcal{Hom}(M,M)$ factors uniquely through a sheaf morphism $\mathbb{C}_X\to\mathcal{Hom}(M,M)$.)