TITLE: When is a diffeomorphism a bundle map?
QUESTION [5 upvotes]: Let $F\rightarrow E_0 \rightarrow B$ and $F\rightarrow E_1\rightarrow B$ be two smooth fiber bundles. Suppose $E_0$ and $E_1$ are diffeomorphic.
What are the obstructions for $E_0$ and $E_1$ to be diffeomorphic via a bundle map?
To be more specific, I am interestid in the following cases:

Vector bundles ($F=\mathbb{R}^n$)
Vector bundles over spheres ($B=S^n$)
Sphere bundles over spheres ($F=S^m$, $B=S^n$)

REPLY [3 votes]: I actually found a partial answer to the case of vector bundles over spheres (that is, under certain dimensional restrictions): https://www.sciencedirect.com/science/article/pii/S0040938399000397