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?



Manifolds distinguished by Gromov-Witten invariants?
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.


