Fusion and elastic scattering cross sections were measured for $^{20}\mathrm{Ne}$ bombardments of $^{40}\mathrm{Ca}$ from 44 to 70 MeV. The evaporation residues from the compound nucleus decay were detected by a gas counter telescope. The fusion data were compared to results for $^{16}\mathrm{O}$ + $^{40}\mathrm{Ca}$ within the framework of the proximity, the Siwek-Wilczynska and Wilczynski, and the folding model potentials. Although the ratio $\frac{{\ensuremath{\sigma}}_{\mathrm{fus}}}{{\ensuremath{\sigma}}_{R}}$ is appreciably lower for $^{20}\mathrm{Ne}$ than for $^{16}\mathrm{O}$, consistent results were obtained for ${\ensuremath{\sigma}}_{\mathrm{fus}}$ within the experimental uncertainties.NUCLEAR REACTIONS $^{20}\mathrm{Ne}$ + $^{40}\mathrm{Ca}$, $E=44.1 \mathrm{to} 70.4$ MeV; Measured evaporation residue $\ensuremath{\sigma}(E,\ensuremath{\theta})$. Deduced $\ensuremath{\sigma}(E)$ total fusion excitation functions. Measured elastic scattering $\ensuremath{\sigma}(\ensuremath{\theta})$. Natural target. Deduced ion-ion potentials.