Any two symplectic forms on $\mathbb{R}^{2n}$ are in the same cohomology class. But the usual symplectic form on a ball of radius 1 in Darboux coordinates does not have the same volume as the usual symplectic form on a ball of radius 2 in Darboux coordinates, even though the rescaling is a diffeomorphism. But maybe you want compact examples.

A much fancier class of examples: Gromov, Pseudoholomorphic curves in symplectic manifolds, p. 313, corollary $0.4.A_2'$ proves that there are symplectic structures on $\mathbb{R}^{2n}$ which do not embed into the usual Darboux symplectic structure.

For compact manifolds, Moser's homotopy method proves that cohomologous symplectic structures are symplectomorphic; see the standard textbooks, like McDuff and Salamon, or Arnol'd.