Very rougly said, the Darboux Theorem states that all symplectic manifolds (of the same finite dimension) "look the same", i.e. their symplectic form could be brought to look like the standard one - $dx_i \land dx_j$...one of the effects is that the symplectic structure doesn't "encode" curvature.
But on the symplectic 2-sphere, the symplectic form is $\omega =r^2sin\theta d\theta \land d\phi$, which is different...
What I am missing?