A function and its tangent map Say we have two smooth manifolds and a map $F$ between them with corresponding tangent map $dF$. 
Next consider the set of curves on this manifold or equivalently
the set of smooth sections of the tangent bundle. 
Now $F$ maps sets i.e images of curves while $dF$ maps the curves via the associated section of the tangent bundle. Hence the two functions $F$ and $dF$ really contains the same information in some sense.
Am I on to some fundamental idea of having tangetspaces with this observation or is it complete nonsense?
 A: Well this is definitely not nonsense, but I would not say they contain the same information. The tangent map is defined locally, so we should really write $dF_p$, and it encodes the infinitesimal information (or linear approximation) about the map $F$ in a neighborhood of the point $p$. Also worthy of note is that while $dF_p$ is always defined locally, one cannot always map a global vector field on a manifold $M$ to a global vector field on a manifold $N$. It can always be done if $F$ is a diffeomorphism, but if the map is not injective, we might end up with two tangent vectors at the same point. This wikipedia page has more info about what the tangent map $dF$ really represents.
Also I don't understand what you mean by the set of curves on a manifold is equivalent to the sections of the tangent bundle. The sections are smooth global vector fields, and while they are intimately related (by integral flows), I would not say they are equivalent. In fact, if you have a curve in a manifold, you can in many cases define different smooth vector fields along that curve.
