Here is an answer to the REFINED question given to me by Richard Thomas.
In this refined version one looks for an example such that the cohomology
classes of two symplectic forms coincide.

In a later paper 1996, Duke Vol. 83
TOPOLOGICAL SIGMA MODEL AND DONALDSON TYPE
INVARIANTS IN GROMOV THEORY, Ruan proved that such refined examples exist.
He says in this paper that for product examples $V\times S^2$
from the paper in JDG 1994
(cited by Mike Usher) he does not know whether the classes of
constructed symplectic forms can coincide as well. In fact this does not seem very plausible.

These refined examples are two $3$-dimensional Calabi-Yau manifolds,
constructed by Mark Gross. The construction is described in the paper
of Mark Gross (1997): "The deformation space of Calabi-Yau $n$-folds with canonical singularities can be obstructed". One $3$-dimensional Calabi-Yau
is a smooth anti-canonical section of $\mathbb CP^1\times \mathbb CP^3$ and the over is a smooth anti-canonical section of the projectivsation of the bundle
$O(-1)+O+O+O(1)$ over $\mathbb CP^1$.

The construction of Gross is recalled on pages 47-48 of
http://xxx.soton.ac.uk/PS_cache/math/pdf/9806/9806111v4.pdf

Using Wall's theorem Ruan proves that these Calabi-Yau
manifolds are differomorphic. Then he studies the quantum cohomology
rings of these Calabi-Yaus and proves that they are different.