How much do characteristic classes fail to characterize bundles? Given a group $G$, let $E \to B$ be a principal $G$-bundle. It is
well-known that when $B$ is a nice enough topological space (e.g.
CW-complex), such a thing corresponds to a connected component of
$Hom(B,BG)$. So say the bundle corresponds to $[f] \in
\pi_0(Hom(B,BG))$. The characteristic classes of the bundle is
therefore the pull-backs $f^\star(\omega)$ where $\omega \in
H^\star(BG;\mathbb{Z})$.
Question: I am interested in to what extent characteristic
classes fail to characterize bundles.
As $B$ varies, I believe the question is equivalent to
Question': To what extent does the cohomology ring
$H^\star(BG;\mathbb{Z})$ fail to characterize $BG$ up to
homotopic equivalence.
It is well-known that taking cohomology does forget much
information. And for good enough spaces still, one needs to
consider the cochain complex as an $E_\infty$ algebra. Moreover,
any (good enough) space $X$ is equivalence to $B\Omega X$, so I
need to restrict my questions more.
Question'': What can we say about Question' for G being the
most popular groups, e.g. finite groups, compact Lie groups,
loop group of $S^1$.. etc.
 A: I am just posting my comment as an answer.  This question comes up fairly often for algebraic geometry students who are learning about moduli spaces of stable vector bundles, particularly in the case that the base projective manifold has dimension one or two.  In both of these cases, if we fix the first Chern class and let the second Chern class increase, the moduli spaces are eventually connected (or empty), reflecting the fact that there are no discrete invariants other than the first and second Chern classes.
However, it is not true that the homotopy-theoretic information of a vector bundle is faithfully encoded by the Chern classes.  Indeed, there are many nonvanishing, odd-degree homotopy groups of classifying space $B\text{GL}_{n}(\mathbb{C})$, cf. the following link.
http://felix.physics.sunysb.edu/~abanov/Teaching/Spring2009/Notes/abanov-cpA1-upload.pdf
Since the cohomology of an odd-dimensional sphere is trivial in even degree $>0$, the Chern classes vanish for every complex vector bundle on an odd-dimensional sphere.
A: For discrete $G$, the classifying space $BG$ carries the same homotopical information as $G$. On the other hand, the group cohomology $H^{\star}(BG;\mathbb Z)$ can vanish for non-trivial $G$, as it is known that acyclic groups exist. (However, note that there are no finite acyclic groups.)
