A fiber bundle is a quadruple
$$(\mathcal{M},\pi,\mathcal{E},\mathcal{F})$$
(where $\mathcal{M},\mathcal{E}$ and $\mathcal{F}$ are topological spaces) with
$$\pi:\mathcal{E}\to\mathcal{M}$$
such that for every $x\in\mathcal{M}$, there exists a $U\subset\mathcal{M}$ with $x\in U$ and an homeomorphism
$$\chi:\pi^{-1}(U)\to U\times\mathcal{F}$$
such that:
$$\chi\circ pr_1=\pi$$
(where $pr_1:\mathcal{M}\times\mathcal{F}\to\mathcal{M}$ is the projection onto the first component, i.e. $\forall (m,f)\in\mathcal{M}\times\mathcal{F}:pr_1(m,f)=m$)
In other words in a fiber bundle what is required is a homeorphism with a cartesian product (locally). Which means that the total space
$\mathcal{E}$ has to be locally homeomorphic to
$\mathcal{M}\times\mathcal{F}$.
In a sheaf, the only requirement is that a topological space
$\mathcal{E}$ is locally homemorphic to another topological space
$\mathcal{X}$, in fact a sheaf is a triple:
$$(\mathcal{E},\pi,\mathcal{X})$$
such that
$\pi$ is a surjective local homeomprhism.