Torelli's theorem states:

Let $R$, $R'$ be compact Riemann surfaces of genus $g$, $J(R)$, $J(R')$ their Jacobian varieties, $\Theta$, $\Theta'$ their respective theta divisors. The Riemann surfaces $R$ and $R'$ are isomorphic if and only if $(J(R), \Theta)$ and $(J(R'), \Theta')$ are isomorphic as principally polarized Abelian varieties.

In this theorem, $J(R)$ and $J(R')$ are required to be isomorphic not only as Abelian varieties but also as principally polarized Abelian varieties. It turns out that the condition for $J(R)$ and $J(R')$ to be isomorphic as Abelian varieties alone need not imply that $R$ and $R'$ are isomorphic.

Where can I find an example that shows that $J(R)$ and $J(R')$ being isomorphic just as Abelian varieties, does not imply that $R$ and $R'$ are isomorphic?