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).

First remark, there is the definition of sheaf from wikipedia (which by the way talks about étalé spaces and that adjunction business) and the équivalent one 1.2. p. 3 of Bredon, Glen E. (1997), "Sheaf theory" which looks much more like that of a bundle (the A in that definition is the étalé space).

The second remark (from this p.2-3) is that a bundle is locally homeomorphic to a cartesian product, whereas a sheaf is locally homeomorphic to the "base space" itself!

Other difference is that a manifold (which a bundle is) is Hausforff, not the étalé space.