TITLE: Vector bundles vs principal $G$-bundles
QUESTION [32 upvotes]: It is well known that a (real) vector bundle $\pi : E\to B$ over a topological space (or manifold) $B$ is a fibre bundle whose fibres
$$F=\pi^{-1}(x), \ \ \ x\in B $$
over any $x\in B$,   are diffeomorphic to a vector space $V$.   On the other hand, a principal $G$-bundle is a fibre bundle $\pi : P\to B$ over $B$ with a right free action of a Lie group $G$ on $P$ such that for any open set $U\subset B$, the locally trivial fibrations defined by:
$$
\Phi_{U} : \pi^{-1}(U)\to U\times G,  \ \ \Phi_{U}(p)=(\pi(p), \varphi_{U}(p)).    
$$
Here
$$
\ \varphi_{U} :\pi^{-1}(U)\to G 
$$
 is a $G$-equivariant map, that is $\varphi_{U}(pg)=\varphi_{U}(p)g$, for all $p\in\pi^{-1}(U)$ and $g\in G$. In the last case the  fibers are submanifolds of $P$ which are always diffeomorphic with the structure group $G$. 
Although for any vector bundle $\pi : E\to B$, its  fibers  $F\cong V$   can be considered as Lie group with operation the vector addition, in general we do not include the vector bundles
as examples of principal $G$-bundles (Although, to every vector bundle we can associate  the frame bundle which is a ${\rm GL}_{n}\mathbb{R}$-principal bundle, but I don't speak here about associated bundles).
My question is about a good explanation about the fact that IN GENERAL vector bundles (themselves) do not give   examples of principal $G$-bundles. For example, the tangent bundle $TM$ of a smooth manifold is a prototype example of a vector bundle, but itself it cannot be considered as a principal bundle for a Lie group $G$, is this true? 
Thus I ask:
Which is the basic difference between a vector bundle an a $G$-bundle, which does not allows us (almost always??), to consider the vector bundles themselves as examples of $G$-bundles?
For example, a $G$-bundle is trivial (isomorphic to the product bundle), if and only if it admits a global section, but I think that this is not true  for vector bundles.
A second question is about  examples  of vector bundles which can be considered the same time as $G$-bundles for some Lie group (I think that such an example is a cylinder)

REPLY [15 votes]: I believe you have answered this yourself. Principal bundles are trivial iff they admit a global smooth section. Vector bundles always admit a global smooth section (zero section). 
Therefore vector principal G-bundles are always trivial. The only available examples of vector G-bundles are thus of the form M x G, where G is both a vector space and a Lie group. Any examples of such G except for Abelian?