The μ-APPJ is a well-investigated atmospheric pressure RF plasma jet. Up to now, it has mainly been operated using helium as feed gas due to stability restrictions. However, the COST-Jet design including precise electrical probes now offers the stability and reproducibility to create equi-operational plasmas in helium as well as in argon. In this publication, we compare fundamental plasma parameters and physical processes inside the COST reference microplasma jet, a capacitively coupled RF atmospheric pressure plasma jet, under operation in argon and in helium. Differences already observable by the naked eye are reflected in differences in the power-voltage characteristic for both gases. Using an electrical model and a power balance, we calculated the electron density and temperature at 0.6 W to be $9\times {10}^{17}\,{{\rm{m}}}^{-3}$, 1.2 eV and $7.8\times {10}^{16}\,{{\rm{m}}}^{-3}$, 1.7 eV for argon and helium, respectively. In case of helium, a considerable part of the discharge power is dissipated in elastic electron-atom collisions, while for argon most of the input power is used for ionization. Phase-resolved optical emission spectroscopy reveals differently pronounced heating mechanisms. Whereas bulk heating is more prominent in argon compared to helium, the opposite trend is observed for sheath heating. This also explains the different behavior observed in the power-voltage characteristics.