First of all, the Vianna tori are distinguished by the Fukaya category. To get an object of the category, you need to pick a torus and a rank 1 local system on it. The set of local systems making the torus into a nonzero object of an invariant which, I think, distinguishes these tori (it's the same as the critical set of the superpotential).
For something which goes beyond the Fukaya category, you could look at Mohammed Abouzaid's result on exotic spheres which uses higher dimensional moduli spaces of holomorphic curves
https://arxiv.org/abs/0812.4781
For a concrete statement about what Fukaya categories cannot see inspired by this result, Georgios Dimitroglou Rizell and I came up with some compactly supported symplectomorphisms of cotangent bundles of spheres which are not Hamiltonian isotopic to the identity through compactly supported things but which act trivially on the Fukaya category.