What's an example of two non-isomorphic vector bundles $E,F$ over the same base such that the total spaces $Total(E), Total(F)$ are homeomorphic? Assume that rank of these bundles is the same as dimension of the base manifold. 

(EDIT: Mike Miller brought up the excellent point that I need to look at bundles which are not isomorphic covered by any self-homeomorphism of the base, otherwise one can look at two codimension $2$ submanifolds on a manifold such that one can be taken to another by an ambient homeomorphism, with different fundamental classes, and then look at the corresponding complex line bundles. This cannot happen for bundles with the rank of the base, so easiest example comes in dimension $3$, say, eg the torus and the line bundles coming from two of the generator circles in $H\_1$)

Notice that the Euler class of the bundle is an obstruction from this happening - if the Euler classes were different, the total spaces would have different (compactly supported) cohomology ring structure (the self-intersection number of the zero sections would be different)

The simplest example I can come up with which passes the Euler class test is $TS^5$ and $S^5 \times \Bbb R^5$. But $TS^5$ seems to have a complicated total space, so it's unclear how to see if they are or are not diffeomorphic (if not, how would one prove that?). 

It seems to me that there is no easy general technique to determine the diffeomorphism type of the total space from the vector bundle once the Euler class is ruled out. So any general comment along the lines of this would be much appreciated, even as answers.