It is well known that the exercise of logic never adds to our knowledge: its role is to make a certain aspect of that knowledge clearer or more explicit, while keeping all the rest conveniently out of our sight.
Tommaso Toffoli, in "Entropy? Honest!". Entropy 18, 247 (2016). doi: 10.3390/e18070247.
The advance of the geometric formulation of Hamiltonian mechanics, and pretty much of any physical theory if you ask me, is "to make a certain aspect of that knowledge clearer or more explicit, while keeping all the rest conveniently out of our sight". As far as I know, there are no systems that you actually need to describe using the geometric formulation (maybe someone with more expertise might agree or disagree with me later), but the geometric formulation does allow you to see some notions from a better angle.
For example, canonical transformations. When formulating Hamiltonian mechanics in terms of symplectic geometry, you notice that the coordinates and momenta you're using to describe your physical system are merely coordinates on a manifold. You could just as well use different choices, in the same way you can choose between using Cartesian or spherical coordinates in Euclidean 3D space. Symplectic transformations are then simply changing coordinates on phase space in a way that the expression of a geometric object known as the symplectic form (AKA the Poisson brackets) is kept invariant. Of course, you can do all of the usual computations without thinking about these things, but knowing of these details allows you to get a different angle of how Mechanics works, and understand it differently and deeper. Some proofs also might get more natural on a more robust mathematical setting. Sometimes, to spend more time on the definitions that go into a theory later allows you to take a lot of shortcuts when proving more complex results.
The framework of quantum mechanics is vastly different from that of classical mechanics. While classical mechanics lives on a (often finite dimensional) symplectic manifold, quantum mechanics lives on a (nearly always infinite dimensional) Hilbert space. The framework of quantum mechanics is not a mere generalization of classical mechanics, it does have some quite different features.
As far as I know, there is difficulty in obtaining a well-defined "quantization" map, since in principle different quantum systems could lead to the same classical theory (naively, consider the quantum Hamiltonian $H = AB - BA$ for some operators $A$ and $B$, which classically would be simply $\mathcal{H} = 0$). On a more mathematical setting, I guess the phrasing goes something like "quantization is not a functor" (see this post on MathOverflow), but I'm mentioning this for mere completeness, I don't really understand what the phrase means. If you're interested in see how quantization can be done for some linear classical systems using the symplectic structure of the classical theory, Chap. 2 of Wald's Quantum Field Theory in Curved Spacetime and Black Hole Thermodynamics has a nice discussion. I think the course notes by Geroch on geometrical quantum mechanics also discuss that (with a review of the geometric formulation of classical mechanics).
I'm not sure whether there is a mathematically well-defined way of uniquely obtaining the symplectic manifold of a classical theory from the quantum theory (although I do hope so hahaha). Perhaps someone better informed in those details might add an answer that is more complete in that direction.
