The non-elliptic case is easier. By a theorem of Bondal-Orlov, see [Huybrechts, Fourier-Mukai transforms in algebraic geometry, Theorem 4.11]:
Theorem. Let $X$ and $Y$ be smooth projective varieties with equivalent derived category $D^b(X)$ and $D^b(Y)$. If the (anti)-canonical bundle of $X$ is ample, then $X$ and $Y$ are isomorphic.
For elliptic curve, we need a generalized theorem by Kawamata, See [Kawamata, D-equivalence and K-equivalence] or [Huybrechts, Fourier-Mukai transforms in algebraic geometry, Theorem 6.15]:
Theorem. Let $X$ and $Y$ be smooth projective varieties with equivalent derived category $D^b(X)$ and $D^b(Y)$. Then the (anti)-canonical bundle of $X$ is nef if and only if the (anti)-canonical bundle of $Y$ is nef.
This theorem implies if $X$ is elliptic curve, the so is $Y$.
By [Huybrechts, Fourier-Mukai transforms in algebraic geometry, Theorem 5.39], the hodge structuces of $X$ and $Y$ are the same, which yeilds that $X$ and $Y$ are isomorphic.