Why is it worth mentioning that something is symplectic?

This question is a little like asking why it's worth mentioning that an electric field is in the room.

As an important characteristic, I'd point out that if you have a symplectic structure, you have a Poisson algebra. That means that the functions $$f:P\in \mathcal M\ \longrightarrow\ f(P)\in\mathbb{R}$$ on your manifold can not only do things like

$$(f,g,h,P)\ \longrightarrow\ f(P)g(P)+h(P),$$

but also

$$(f,g,P)\ \longrightarrow\ \{f,g\}(P).$$

Consequently, if you add a symplectic structure in your function algebra, some awesome results occur. Notice that the structure as well as the manifold you consider might be wild, but the Possion bracket has some qualities to it, which are true in general.

Why is it worth mentioning that something is symplectic?

This question is a little like asking why it's worth mentioning that an electric field is in the room.

As an important characteristic, I'd point out that if you have a symplectic structure, you have a Poisson algebra. That means that not only can the functions $$f:P\in \mathcal M\ \longrightarrow\ f(P)\in\mathbb{R}$$ on your manifold do things like

$$(f,g,h,P)\ \longrightarrow\ f(P)g(P)+h(P),$$

but also like

$$(f,g,P)\ \longrightarrow\ \{f,g\}(P).$$

Consequently, if you add a symplectic structure in your function algebra, some awesome results occur. Notice that the structure as well as the manifold you consider might be wild, but the Possion bracket has some qualities to it, which are true in general.