We investigate systematically the spin-Nernst effect in Neel and zigzag ordered honeycomb antiferromagnets. Monolayers of transition-metal trichalcogenides, MnPSe3, MnPS3, and VPS3 show an antiferromagnetic Neel order while CrSiTe3, NiPS3, and NiPSe3 show an antiferromagnetic zigzag order. We extract the exchange and Dzyaloshinskii-Moriya interaction parameters from ab initio calculations. Using these parameters, we predict that the spin-Nernst coefficient is at least two orders of magnitude larger in zigzag compared to the Neel ordered antiferromagnets. We find that this enhancement relies on the large band splitting due to the symmetry of magnetic configuration in the zigzag order. Our calculations indicate that the Dzyaloshinskii-Moriya interaction is the underlying factor for the spin-Nernst effect in both cases, although with different microscopic mechanisms. In the case of Neel antiferromagnets, magnon bands already possess a Berry curvature and introducing the Dzyaloshinskii-Moriya interaction splits the magnon bands with the opposite helicity throughout the Brillouin zone which results in an unbalanced population of magnons carrying opposite spins. In the case of zigzag antiferromagnets, magnon bands do not possess the Berry curvature but they are split for opposite helicity magnons due to symmetry of the system. In this case, introducing the Dzyaloshinskii-Moriya interaction induces the Berry curvature and results in the spin-Nernst effect. Due to large magnon band splitting, the spin-Nernst effect in zigzag antiferromagnets is stronger than Neel antiferromagnets.