Does Fukaya see all symplectic topology?

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 is 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.

http://arxiv.org/abs/1407.3173

This answer just provides some general comments pertatining to the question asked in the title of the post. (When I first started graduate school, I was somewhat skeptical of Fukaya categories, and in fact went around asking people when one could show that Fukaya catgories can actually distinguish Lagrangians up to Hamiltonian isotopy. So I am quite sympathetic to the question.)

It is difficult to give a precise answer to how much symplectic topology Fukaya categories know, because there are many flavors of `the Fukaya category' and it seems likely that more will be created in the future. After all, Fukaya categories are a particular approach to packaging holomorphic curve counts in a way that is particularly amenable to homological algebra, and there are a lot of interesting curves that one may try to count!

The Vianna tori are actually an excellent example of Fukaya categories influencing more normal symplectic topology; I think I was once told that various people had tried to prove that all tori are isotopic to the clifford or chekanov torus until Vianna came up with his examples, which were, as far as I understand, motivated by mirror symmetry considerations. Vianna does not need the Fukaya category to distinguish his tori, but the superpotential, which is the invariant that Vianna used to distinguish his tori (and which Jonny Evans mentioned), was constructed by FOOO in their book on Fukaya categories -- it is part of the data of the curved Fukaya category. (I don't know when superpotentials of Lagrangian tori were first discussed and computed.) As discussed in the comments to Evans' answer, after uncurving the objects supported on these different tori often isomorphic in the Fukaya category over the Novikov field; but the curved Fukaya category is a core part of the Fukaya-categorical approach to symplectic topology.

It seems worth noting that in some sense the Fukaya category cannot even distinguish discern classical topology. For example, there are various flavors of Fukaya category that one can associate to a Lioville domain $M$, and most of them are zero if $M$ is a subcritical Weinstein domain. Yet, there are many subcritical weinstein domains which have non-symplectomorphic completions; for example, they may have different singular homologies; so the Fukaya category doesn't know about singular cohomology! However, subcritical Weinstein domains satisfy certain h-principles, and so one might say that they have no symplectic topology beyond their smooth topology. The question the boundary between flexibility (e.g. Weinstein domains satisfying an h-principle) and rigidity (Weinstein domains having interesting invariants which preclude them from satisfying h-principles) has long been a major strand of research and has been intertwined with improvements in Fukaya-categorical technology for at least a decade. For example Abouzaid-Seidel's "homologous recombination" paper used Fukaya categories as a very rich source of exotic examples of this sort, and there is later work involving "bulk deformed Fukaya categories" which can detect non-flexibility even when more usual Fukaya categories are empty.

Going to the other extreme, from the perspective of symplectic geometry, the symplectic ball of symplectic form $\omega$ is not the same as the symplectic ball with symplectic form $2\omega$; and indeed, ball-packing problems (e.g. how much of the symplectic ball in $C^n$ of radius 1 can I fill by $k$ equal-radius symplectic balls embedded disjointly via symplectomorphisms) are a classicl problem that is sensitive to this difference. There are some fukaya-categorical invariants which are also sensitive to phenomeon, which goes "beyond symplectic topology"; for example, one might say that symplectic cohomology over the novikov ring is part of the general structure of the fukaya category, and this object gives rise to many interesting quantitative invariants of a Liouville domain. (But I think that in the Fukaya category over the novikov ring, branes supported on Hamiltonian-isotopic Lagrangian submanifolds are not isomorphic, because the continuation maps have positive valuation; so one might complain that this invariant sees too much!). For more of this flavor, you might look at Kyler Siegel's recent paper on `higher symplectic capacities', which use some of the higher-algebraic structure on filtered symplectic cohomology, which is closely related to symplectic homology over the novikov ring, to build interesting quantitative invariants of Lioville domains. The understanding of what higher-algebraic structure exists on symplectic cohomology has been closely tied to the development of the theory of Fukaya categories. But note that Siegel uses the formalism of symplectic field theory, which is intimately related to the formalism of Fukaya categories in a way that is still being unraveled.

Yet, going back to the original statement about smooth topology, by being cleverer with holomorphic curves one can construct invariants which are tricky to extract from any Fukaya category. (Evans already mentioned several such invariants, but I think the following example is fun because it is so close to being of Fukaya-categorical flavor.) For example, Jingyu Zhao in `Periodic Symplectic Cohomologies' constructs a version of "Periodic Symplectic Cohomologies" which satisfies a localization theorem; namely, it is just the homology of the underlying manifold tensored with $\mathbb{Q}[[u]][u^-1]$, via a nontrivial isomorphism. One reason this is interesting because, as described two paragraphs ago, the flavors of Fukaya category that are typically studied for a Weinstein domain cannot possibly know about the homology of the Weinstein domain. Yet, Zhao's invariant is not a quantitative invariant, in the sense that it is an invariant of the Lioville domain after completion, so it's the same "kind" of invariant as the wrapped fukaya category. There is a variant of "periodic symplectic cohomology" that one can compute using the usual wrapped fukaya category of a Weinstein domain by some homological algebra, see Ganatra's preprint on the cyclic open-closed map, but it does not satisfy the localization theorem.

Anyway, we don't know how much symplectic topology Fukaya categories see, but the Fukaya categorical perspective has given rise to an immense number of invariants that see ever more symplectic topology. One must just keep asking "how much symplectic topology can be seen"? Go forth and classify Lagrangian tori in $CP^3$ up to Hamiltonian isotopy using any tools you have!