I'd like to add two more examples.
The first one is in the negative direction. Let $\pi:E^{2n}\rightarrow\mathbb{C}$ be an exact Lefschetz fibration, by the work of Seidel one can associate its Fukaya category $\mathscr{F}(\pi)$, which can be realized as a directed $A_\infty$ algebra or a diagonal bimodule. Denote by $\mathscr{F}(\pi)^\vee$ its dual bimodule. Seidel constructed the following bimodule map
$\beta:\mathscr{F}(\pi)^\vee[-n]\rightarrow\mathscr{F}(\pi)$
which describes the natural transformation of degree $n$ from the Serre functor to the identity. Its an observation due to Maydanskiy that there are examples of exact Lefschetz fibrations with quasi-isomorphic $\mathscr{F}(\pi)$, but with different bimodule maps $\beta$. This shows that one should include $\beta$ as an additional piece of information to distinguish exact symplectic manifolds which admit Lefschetz fibrations.
The second example is in the positive direction. Weiwei Wu proved the following:
The special isogenous tori are symplectomorphic if and only if they have equivalent derived Fukaya categories.
A special isogenous torus, by definition, is obtained by taking products of tori which are finite group quotients of a standard product torus. The proof is a combination of the result of Abouzaid-Smith on homological mirror symmetry for standard $T^{2n}$ and finite group actions on Fukaya categories which generalize the case of a $\mathbb{Z}/2$-action considered by Seidel in the definition of $\mathscr{F}(\pi)$.
This result is also expected to be true for symplectic tori equipped with linear symplectic forms.