You write:



<Blockquote> So, I'm wondering for some applications where I really need to use a bundle --- where some important fact is not true for general submersions (or, surjective submersions with connected fibers, say).

</Blockquote>

Actually, I am going to play devil's advocate here: *sometimes it's better to have 
a submersion!* This point comes up in a very relevant way in the classical smoothing theory of topological manifolds.  Siebenmann (cf. Kirby and Siebenmann's book) defines a moduli space of smoothings of a topological manifold $M$ to be the space of $$(N,f)$$ such that $N$ is smooth and $f: N \to M$ is a homeomorphism.

Siebenmann chooses to topologize this in what seems a funny way: a $k$-simplex of such things is a pair $(N,f)$, where now $N \to \Delta^k$ is a smooth submersion 
(*not necessarily proper if $M$ isn't compact!*) and $f: N \to M \times \Delta^k$ is a homeomorphism which is compatible with projection to $\Delta^k$. This gives a $\Delta$-space (a simplicial set w/o degeneracies). Call its geometric realization $\text{Sm}(M)$.


*Why doesn't he just topologize families as fiber bundles?*


Here's why:

Let ${\cal O}_M$ be the poset of open subsets of $M$ which are abstractly homeomorphic to open balls.
The fundamental theorem of smoothing theory asserts that the contravariant functor
$\text{Sm} : {\cal O}_M \to \text{Top}$ given by
$$
U \mapsto \text{Sm}(U)
$$
is a "homotopy sheaf" if $\dim M \ge 5$, i.e., the (restriction) map
$$
\text{Sm}(M) \to \underset{U \in {\cal O}_M} {\text{holim}}\quad  \text{Sm}(U)
$$
is a homotopy equivalence. This would not be the case if we had defined the families
as bundles (rather than as submersions). Note: we cannot appeal to Ehresmann here as 
the submersions which are used in the define $k$-simplices in $\text{Sm}(U)$ are not assumed to be proper.