Manifolds distinguished by Gromov-Witten invariants? What is a simplest example of a manifold $M^{2n}$ that admits two symplectic structures with isotopic almost complex structures, and such that  Gromov-Witten invariants of these symplectic structures are different? 
(unfortunately I don't know any example...) If we don't impose the condition that almost complex structures are isotopic, such examples exist in dimension 6. 
Added
Refined question. Is there a manifold $M^{2n}$ with two symplectic forms $\omega_1$, $\omega_2$,
such that the cohomology classes of $\omega_1$ and $\omega_2$ are the same and
the corresponding almost complex structures are homotopic, 
but at the same time the Gromov-Witten invariants are different?
 A: The examples in Ruan's paper "Symplectic topology on algebraic 3-folds" (JDG 1994) seem to qualify: take any two algebraic surfaces V and W which are homeomorphic but such that V is minimal and W isn't.  These are nondiffeomorphic, but VxS2 and WxS2 are diffeomorphic, and Ruan gives lots of examples (starting with V equal to the Barlow surface and W equal to the 8-point blowup of CP2) where the diffeomorphism can be arranged to intertwine the first Chern classes, whence by a theorem of Wall the almost complex structures are isotopic. However, the distinction between the GW invariants between V and W (which holds because V is minimal and W isn't) survives to VxS2 and WxS2, so VxS2 and WxS2 aren't symplectic deformation equivalent.
A: 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.
