The mass difference between the ${\mathrm{\ensuremath{\Omega}}}_{b}^{\ensuremath{-}}$ and ${\mathrm{\ensuremath{\Xi}}}_{b}^{\ensuremath{-}}$ baryons is measured using proton-proton collision data collected by the LHCb experiment, corresponding to an integrated luminosity of $9\text{ }\text{ }{\mathrm{fb}}^{\ensuremath{-}1}$, and is found to be $m({\mathrm{\ensuremath{\Omega}}}_{b}^{\ensuremath{-}})\ensuremath{-}m({\mathrm{\ensuremath{\Xi}}}_{b}^{\ensuremath{-}})=248.54\ifmmode\pm\else\textpm\fi{}0.51(\mathrm{stat})\ifmmode\pm\else\textpm\fi{}0.38(\mathrm{syst})\text{ }\text{ }\mathrm{MeV}/{c}^{2}$. The mass of the ${\mathrm{\ensuremath{\Omega}}}_{b}^{\ensuremath{-}}$ baryon is measured to be $m({\mathrm{\ensuremath{\Omega}}}_{b}^{\ensuremath{-}})=6045.9\ifmmode\pm\else\textpm\fi{}0.5(\mathrm{stat})\ifmmode\pm\else\textpm\fi{}0.6(\mathrm{syst})\text{ }\text{ }\mathrm{MeV}/{c}^{2}$. This is the most precise determination of the ${\mathrm{\ensuremath{\Omega}}}_{b}^{\ensuremath{-}}$ mass to date. In addition, the production rate of ${\mathrm{\ensuremath{\Omega}}}_{b}^{\ensuremath{-}}$ baryons relative to that of ${\mathrm{\ensuremath{\Xi}}}_{b}^{\ensuremath{-}}$ baryons is measured for the first time in $pp$ collisions, using an LHCb dataset collected at a center-of-mass energy of 13 TeV and corresponding to an integrated luminosity of $6\text{ }\text{ }{\mathrm{fb}}^{\ensuremath{-}1}$. Reconstructing beauty baryons in the kinematic region $2<\ensuremath{\eta}<6$ and ${p}_{T}<20\text{ }\text{ }\mathrm{GeV}/c$ with their decays to a $J/\ensuremath{\psi}$ meson and a hyperon, the ratio $\frac{{f}_{{\mathrm{\ensuremath{\Omega}}}_{b}^{\ensuremath{-}}}}{{f}_{{\mathrm{\ensuremath{\Xi}}}_{b}^{\ensuremath{-}}}}\ifmmode\times\else\texttimes\fi{}\frac{\mathcal{B}({\mathrm{\ensuremath{\Omega}}}_{b}^{\ensuremath{-}}\ensuremath{\rightarrow}J/\ensuremath{\psi}{\mathrm{\ensuremath{\Omega}}}^{\ensuremath{-}})}{\mathcal{B}({\mathrm{\ensuremath{\Xi}}}_{b}^{\ensuremath{-}}\ensuremath{\rightarrow}J/\ensuremath{\psi}{\mathrm{\ensuremath{\Xi}}}^{\ensuremath{-}})}=0.120\ifmmode\pm\else\textpm\fi{}0.008(\mathrm{stat})\ifmmode\pm\else\textpm\fi{}0.008(\mathrm{syst})$, is obtained, where ${f}_{{\mathrm{\ensuremath{\Omega}}}_{b}^{\ensuremath{-}}}$ and ${f}_{{\mathrm{\ensuremath{\Xi}}}_{b}^{\ensuremath{-}}}$ are the fragmentation fractions of $b$ quarks into ${\mathrm{\ensuremath{\Omega}}}_{b}^{\ensuremath{-}}$ and ${\mathrm{\ensuremath{\Xi}}}_{b}^{\ensuremath{-}}$ baryons, respectively, and $\mathcal{B}$ represents the branching fractions of their respective decays.