What are the differences between a fiber bundle and a sheaf? They are similar. Both contain a projection map and one can define sections, moreover the fiber of the fiber bundle is just like the stalk of the sheaf.

But what are the differences between them?

Maybe a sheaf is more abstract and can break down, while a fibre bundle is more geometric and must keep itself continuous. Any other differences?
If $(X,\mathcal{O}_X)$ is a ringed topological space, you can look at locally free sheaves of $\mathcal{O}_X$-modules on $X$.

If $\mathcal{O}_X$ is the sheaf of continuous functions on a topological manifold (=Hausdorff and locally homeomorphic to $\mathbb{R}^n$), or the sheaf of smooth functions on a smooth manifold, you get fiber bundles (the sheaf associated to a fiber bundle is the sheaf of "regular" (=continuous or smooth here) sections).
"In the geometric context the cast of characters changes: on the Galois side we now have vector bundles with flat connection on a complex Riemann surface X in the global case, and on the punctured disc D around a point of X in the local case. The definition of the objects on the other side of the geometric Langlands correspondence is more subtle. It is relatively well understood (after works of A. Beilinson, V. Drinfeld, G. Laumon and others) in the special case when the flat connection on our bundle has no singularities. Then the corresponding objects are the so-called “Hecke eigensheaves” on the moduli spaces of vector bundles on X. These are the geometric analogues of unramified automorphic functions. The unramified global geometric Langlands correspondence is then supposed to assign to a flat connection on our bundle (without singularities) a Hecke eigensheaf." https://math.berkeley.edu/~frenkel/loop.pdf