What does symplecticity imply?

Standard Hamiltonian mechanics in $N$-particle phase space $R^{6N}$ is inadequate to describe mechanical systems of interest that are not of the $N$ particle form, for example rigid bodies. However, all major techniques in classical mechanics do not depend on the specific structure of $R^{6N}$ but only on the fact that one can define on it a Poisson bracket.

Thus classical mechanics generalizes without difficulties to mechanics in Poisson manifolds. These are phase spaces on whose smooth function algebra one can define a Poisson bracket with the properties familar from $N$-particle phase space. (For example, the phase space of rigid bodies is the Lie-Poisson manifold of the Lie algebra generating the group of rigid motions.) For conservative classical mechanics in terms of Poisson manifolds see the book
J.E. Marsden and T.S. Ratiu, Introduction to mechanics and symmetry: a basic exposition of classical mechanical systems, Springer 1999.
\url{http://higherintellect.info/texts/science_and_technology/physics/Introduction to Mechanics and Symmetry.pdf}

An important class of Poisson manifolds are the symplectic manifolds, where the Poisson bracket is defined though a symplectic form. (A typical example is the cotangent space of a configuration manifold.) The importance of symplectic manifolds stems from the fact that Poisson manifolds typically foliate into symplectic leaves, and any Hamiltonian dynamics restricted to such a leaf is symplectic.

Edit: Although classical mechanics in tectbooks is usually confined to the conservative case, one can add dissipative terms to an otherwise Hamiltonian mechanics. For example, dissipative classical mechanics for realistic fluids is discussed in terms of Poisson brackets in the book
A.N. Beris and B.J. Edwards, Thermodynamics of flowing systems with internal microstructure, New York 1994

Classical mechanics is the study of second-order systems. The obvious geometric formulation is via semi-sprays, ie second-order vectorfields on the tangent bundle. However, that's not particularly useful as there's no natural way to derive a semi-spray from a function (ie potential).

Lagrangian and Hamiltonian mechanics are two solutions to that problem. While these formalisms are traditionally formulated on the tangent and cotangent bundles (ie velocity and momentum phase space), they were further generalized: Lagrangian mechanics led to the jet-bundle formulation of classical field theory, and Hamiltonian mechanics to the Poisson structure.

The symplectic structure is a stripped-down version of the structure of the cotangent bundle - the part that turned out to be necessary for further results, most prominently probably phase space reduction via symmetries. It doesn't feature prominently in undergraduate mechanics lecture (at least not the ones I attended) because when working in canonical coordinates, it takes a particular simple form - basically the minus in Hamilton's equations, where it's used similarly to the metric tensor in relativity, ie to make a contravariant vector field from the covariant differential of the Hamilton function.

Symplectic geometry also plays its role in thermodynamics: As I understand it, the Gibbs-Duhem relation basically tells us that we're dealing with a Lagrangian submanifold of a symplectic space, which is the reason why the thermodynamical potentials are related via Legendre transformations.

Symplectic geometry is may be the cornerstone of the geometrization of physics. In addition to the very known fact that classical mechanics can be described by symplectic geometry, given some other structures, symplectic spaces can be quantized to produce quantum mechanics as well. A subclass of symplectic geometries namely Kaehler geometry is especially important to quantization problems.

Many physical theories such as Yang-Mills and gravity have descriptions in the context of symplectic geometry, please see the review: THE SYMPLECTIZATION OF SCIENCE by Gotay and Isenberg.

Also many types of dissipative systems can be treated using symplectic geometry if we allow complex Hamiltonians please see S.G. Rajeev's article.

Finally, I want to remark that in the symplectic geometry terminology there is a distinction between symplectic and Hamiltonian vector fields, while a symplectic vector field is required to leave the symplectic structure invariant, a Hamiltonian vector field is required in addition to produce an exact form upon the contraction with the symplectic form. For example the vector fields along the generators of the two-torus are symplectic but not hamiltonian. This distinction exists only if the symplectic manifold is nonsimply connected.