I'm in the process of re-learning some differential geometry after a couple years of not doing math professionally and I'm finding myself stuck on what's probably a pretty simple point. I'm going to spell out where I think I am right now; please tell me if something is wrong. For simplicity, let's say $G$ is a Lie group and all the spaces we're talking about are smooth manifolds. Fix forever a manifold $F$ and a faithful action of $G$ on $F$.
Suppose we're given a principal $G$-bundle $P$ over some manifold $B$. We can form an associated "fiber bundle with fiber $F$ and structure group $G$" by taking $P\times_GF=(P\times F)/\sim$, where $(p.g,f)\sim(p,g.f)$.
Alternatively, you can see whether a fiber bundle $E$ arises in this way by asking whether there is a family of trivializations of $E$ for which the corresponding transition functions can be taken to land in $G$.
The question is this: is "having structure group $G$" just a property of fiber bundles, or is a "fiber bundle with structure group $G$" a fiber bundle with some extra structure? If the latter is true, is there a nice description of what this extra structure is that doesn't depend on a family of trivializations? Is it just a chosen isomorphism of $E$ with $P\times_GF$ for some $P$? If this is true, can there be two nonisomorphic principal bundles $P,P'$ for which $P\times_GF$ and $P'\times_GF$ (for the same action on $F$) are isomorphic as (ordinary, non-$G$) fiber bundles?
The example you must have in mind is an $n$-dimensional manifold $M$ and its tangent bundle $TM$. $TM$ can be obtained from the bundle of frames and $\mathbb{R}^n$ where $Gl(n)$ acts naturally on $\mathbb{R}^n$.
There is a notion of $G$-structure defined on $M$ $G\subset Gl(n)$; this is equivalent to saying that $TM$ has a $G$-reduction; that is the coordinate change of $TM$ tak their values in $G$, a Riemannian manifold is an $O(n)$-reduction. Definitively, there can exists two $G$-structures defined on the same manifold which are not isomorphic.