Differences between locally free sheaves and local systems

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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?