We show that the moduli space $\overline{M}_{X}(\nu)$ of Gieseker stable sheaves on a smooth cubic threefold $X$ with Chern character $$\nu = (3,−H,−H^2 /2,H^3 /6)$$ is smooth and of dimension four. Moreover, the Abel-Jacobi map to the intermediate Jacobian of $X$ maps it birationally onto the theta divisor $\Theta$, contracting only a copy of $$X \subset \overline{M}_X(\nu)$$ to the singular point $0 \in \Theta$. We use this result to give a new proof of a categorical version of the Torelli theorem for cubic threefolds, which says that $X$ can be recovered from its Kuznetsov component $$\text{Ku}(X) \subset D^b(X).$$ Similarly, this leads to a new proof of the description of the singularity of the theta divisor, and thus of the classical Torelli theorem for cubic threefolds, i.e., that $X$ can be recovered from its intermediate Jacobian.