First I apologize because this is not a research question, but I can't get any answer on MathStackExchange...

Let $\pi \colon E \to B$ and $\pi' \colon E' \to B$ two topological fiber bundles on the same base $B$. A $B$-morphism $f \colon E \to E'$ is an isomorphism of fiber bundles iff for all $b \in B$ the induced morphism between fibers $f_{E_b} \colon E_b \to E'_b$ is a homeomorphism.

Is this true ? If not in general, under which additional conditions on the spaces does it become true ?

The direct implication is obvious, but I have huge doubts on the reverse. It is clear that $f$ is a bijection, continuous by definition, so we just have to show that $f^{-1}$ is also continuous. Continuity being a local property we can suppose that both fibrations are trivial, i.e. $f \colon B \times F \to B \times F'$ is of the form $(b,y) \mapsto (b,f_b(y))$ where $f_b \colon F \to F'$ is a homeomorphism for all $b \in B$.

Define $f^{-1} \colon B \times F' \to B \times F$ by $(b,y') \mapsto (b,(f_b)^{-1}(y'))$. Since $f$ is continuous, the coordinate map $(b,y) \mapsto f_b(y)$ is also continuous, and if we can conclude from this that the coordinate map $(b,y') \mapsto (f_b)^{-1}(y')$ is also continuous, $f^{-1}$ is continuous and we are done.

But why should this last step be correct ?

It is correct for vector bundles (F and F' being vector spaces of the same dimension) because in this case $f_b \in L(F,F')$ forall $b \in B$, and $f_b$ can be represented by a matrix whose coefficients are continuous fonctions of $b$. Then the polynomial formula for the inverse matrix shows that the coefficients of $(f_b)^{-1}$ are also continuous.

It is also trivially true with principal bundles because there any $B$-morphism is an isomorphism of principal bundles.

But quid in the general topological case ? Forall $b \in B$, $f_b \in Homeo(F,F')$. With the compact-open topology on this set, can we say that the application $b \mapsto f_b$ is continuous (I hope so) ? And how can we say anything about the map $b \mapsto (f_b)^{-1}$ and $f^{-1}$ ?