Assume we have a fiber bundle $F\to E\stackrel \pi\to B$.
In the wikipedia article it is stated that $E$ the vertical bundle $V=\ker d\pi$ consisting of vectors along the fibers is canonically defined while the horizontal bundle of vectors along the base is not.
Specifying a horizintal subspace of $TE$ is then called an Ehresmann connection on $E$.
But isn't $\pi^* TB$ a canonical subbundle of $TE$ which consists of vectors along $B$?
On
No! There is no reason why $\pi^*TB$ is a subbundle of $TE$ -- this is precisely why you need a connexion.
Recall $\pi^*TB$ is actually constructed as $E\times_B TB$: $$ \pi^*TB=\{(e,(p,v))\in E\times TB\mid \pi(e)=\operatorname{proj}_{TB\to B}(p,v)=p\} $$ where $\operatorname{proj}_{TB\to B}$ is the usual projection $TB\to B$.
By definition of the pullback, $\pi^*B=\{(x,y,v):y\in\pi^{-1}(x), v\in T_xB\}$ therefore it is not a subbundle of $TE$.