I've started reading Hindry-Silverman's Diophantine Geometry and I got a little ahead of myself. Non-isomorphic elliptic curves have non-isomorphic Jacobians, that is because elliptic curves are isomorphic to their Jacobians. However, in the case of two non-isomorphic hyperelliptic curves, are their Jacobians non-isomorphic?
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As said @Matt in the comments, two (projective smooth... ) curves with isomorphic polarized jacobians are isomorphic.
But if their jacobians are isomorphic just as abstract abelian varieties, then the answer is no. See eg. E. Howe: Infinite families of pairs of curves over Q with isomorphic Jacobians, for a recent account.