The question is simple: are Whitney sum bundle and direct product bundle the same? When are they different?
PS: I have realized my mistake: let $E_1 \to B$ and $E_2 \to B$ be two bundles, their Whitney sum is $E_1 \oplus E_2 \to B$ whereas their direct product is $E_1 \times E_2 \to B \times B$. Although $E_1 \oplus E_2 \cong E_1 \times E_2$ their base spaces are different!
Request: I have flagged this question requesting to delete it. Could some moderator help me?
No, they are not the same.
Given two vector bundles $E_1$, $E_2$ over the same base $X$, their Whitney sum is the vector bundle $E_1\oplus E_2$ over $X$.
Given a vector bundle $E$ over $X$ and a vector bundle $F$ over $Y$, their product is the vector bundle $E\times F$ over $X\times Y$. More precisely, $E\times F = p_1^*E\oplus p_2^*F$ where $p_1, p_2$ are the projections onto the first and second factors respectively.