In the first part of the paper we complete the classification of the arithmetical Cohen-Macaulay vector bundles of rank 2 on a smooth prime Fano threefold. In the second part, we study some moduli spaces of these vector bundles, using the decomposition of the derived category of $X$ provided by Kuznetsov, when the genus of $X$ is 7 or 9. This allows to prove that such moduli spaces are birational to Brill-Noether varieties of vector bundles on a smooth projective curve $\Gamma$. When the second Chern class is low we are able to give a more precise description of the moduli space of rank-2 semistable sheaves with fixed Chern classes $\mathbf{M}_{X}(2, c_{1}, c_{2})$. If $g = 7$, we show that the moduli space $\mathbf{M}_{X}(2, 1, 6)$ is isomorphic to a smooth irreducible Brill-Noether variety of dimension 3. Moreover the set of vector bundles contained in $\mathbf{M}_{X}(2, 0, 4)$ is smooth irreducible of dimension 5. If $g = 9$, we prove that $\mathbf{M}_{X}(2, 1, 7)$ is isomorphic to the blow-up of $\operatorname{Pic}(\Gamma)$, where $\Gamma$ is a plane smooth quartic. If $g = 12$, an open set of $\mathbf{M}_{X}(2, 1, d)$ can be described as a quotient with respect to the action of a semisimple group in terms of monads.