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We investigate the mean locally measured velocity dispersions of ionized gas (σgas) and stars (σ*) for 1090 galaxies with stellar masses $\log \, (M_{\!\ast }/M_{\odot }) \ge 9.5$ from the SAMI Galaxy Survey. For star-forming galaxies, σ* tends to be larger than σgas, suggesting that stars are in general dynamically hotter than the ionized gas (asymmetric drift). The difference between σgas and σ* (Δσ) correlates with various galaxy properties. We establish that the strongest correlation of Δσ is with beam smearing, which inflates σgas more than σ*, introducing a dependence of Δσ on both the effective radius relative to the point spread function and velocity gradients. The second strongest correlation is with the contribution of active galactic nuclei (AGN) (or evolved stars) to the ionized gas emission, implying that the gas velocity dispersion is strongly affected by the power source. In contrast, using the velocity dispersion measured from integrated spectra (σap) results in less correlation between the aperture-based Δσ (Δσap) and the power source. This suggests that the AGN (or old stars) dynamically heat the gas without causing significant deviations from dynamical equilibrium. Although the variation of Δσap is much smaller than that of Δσ, a correlation between Δσap and gas velocity gradient is still detected, implying that there is a small bias in dynamical masses derived from stellar and ionized gas velocity dispersions.