When dealing with (locally trivial and smooth) bundles $\pi:E\rightarrow M$ in differential geometry, one speaks about local trivializations or cocycles, but it seems to me the topology of $E$ and $M$ are forgotten. How does it matter?
For example, is there a example such that the topology of $E$ as a manifold may be different from the topology induced by the bundle structure?
The topology (and smooth structure) of $E$ and $M$ are not ignored. A local trivialization of a smooth $F$-bundle $\pi:E\to M$ over an open subset $U\subseteq M$ is a diffeomorphism $\pi^{-1}(U)\to U\times F$ which commutes with the maps to $U$. To say what it means for this to be a diffeomorphism, you use the smooth manifold structure on $U$ (as an open subset of $M$) and also on $\pi^{-1}(U)$ (as an open subset of $E$).