Is every local system with fiber a vector space a locally free sheaf? What are the main differences between these two concepts?
I was playing with the sheaf of sections of $Mo \to S^1$ ($Mo=$Möbius strip) but I don't know if in this case I have to think about a locally free sheaf or a local system.
I was told here that the category of local systems is equivalent to the category of covering spaces then the sheaf of sections described above cannot be a local system right?
Let $M$ be a smooth manifold (the general answer will be similar albeit less simple and more technical in other context e.g. complex analytic/algebraic etc..). The relation between locally free sheaves and local system is not so complicated but could be easy to misinterpret since both exist to model different notions of "a family of vector spaces".
Let $Loc(M) \subset Sh(M)$ be the full subcategory of sheaves on $M$ whose objects are (finite dimensional) local systems. Meaning sheaves for which the exists a cover of $M$ s.t. over each open set in the cover they are isomorphic to a trivial sheaf of a fixed )finite) rank. This is an abelian subcategory (in fact it is closed under extensions as well). To see this just recall that taking stalks is exact.
Let $LocFree(\mathcal{O}_M) \subset \mathcal{O}_M-Mod$ be the full subcategory of $\mathcal{O}_M$-modules on the locally free sheaves (finite rank). Those $\mathcal{O}_M$-modules which are locally isomorphic to free module of a fixed rank. This is not an abelian category! Recall (serre-swan theorem) that this category is equivalent to just the category of vector bundles over $M$.
The relationship between these categories is as follows:
There's always a functor $- \otimes_{k_M} \mathcal{O}_M: LocSys(M) \to LocFree(\mathcal{O}_M)$ (which is not fully faithful!) its essential image consists of those vector bundles which admit a flat connection (the flat connection can be glued from the local trivialization of the local system). This functor induces an equivalence between the categories of $LocSys(M)$ and and vector bundles with flat connections $FlatVect(M)$ (these are vector bundles with additional data and are again a subcategory of $LocFree(\mathcal{O}_M)$ which is not full). The functor in the other direction is given by taking the sheaf of flat sections (which will be a local system).
The sheaf of sections of a vector bundle is a locally free sheaves and not a local system! So in your particular example the sheaf of sections is locally free and the local system you want to associate with the mobius strip is the sheaf of flat sections for some flat connection on the mobius strip. Then you would get the local system corresponding to the non-trivial 2-cover of the circle.