In various settings (geometry, automorphic forms) the symplectic group does appear. In my mind, it may be realized formally as the group of matrices $$GSp(4) = \left\{ g \in GL(4) \ : \ g^T J g = \lambda_g J, \lambda_g \in GL(1) \right\}$$ where $J$ is the block matrix $$J = \begin{pmatrix} 0 & -I_2 \\ I_2 & 0 \end{pmatrix}$$
I understand this is a group and this may be interesting in this respect. But why does it seem so important? I understand it is one of the main types of reductive groups, but is there an intuition behind it? Can we understand geometrically why it is so important? why it is also so genuinely different from $GL(n)$?
Any insight, reference, examples, etc. will help.
There are 3 families of classical groups: linear, orthogonal and symplectic. You know the linear groups, like SL and GL. The orthogonal groups, like O and SO, are essentially rotation groups--they preserve quadratic forms. Symplectic groups preserve sympletic forms. So they are all some sort of automorphism group of basic geometries.
In rank 1, all of these families essentially coincide: $SL(2) = Sp(2)$ and $PGL(2) = SO(3)$ (split forms). These all "fit in" GL(2): SL(2) and Sp(2) as subgroups and SO(3) as a quotient.
In rank 2, there is still some coincidence: you have SL(3), Sp(4), SO(4) and SO(5). SO(4) is basically 2 copies of SL(2), and Sp(4) and SO(5) are closely related, like Sp(2) and SO(3) are. They both "fit in" GSp(4): Sp(4) is a subgroup of GSp(4), and SO(5) = PGSp(4).
Thus GSp(4) "contains" the first cases of non-linear classical groups. It's also in some sense the next closest thing to GL(2) in higher rank. In the Langlands program, it's its own dual group, and contains the theory of Siegel modular forms of degree 2.
See also this MO post.