We present a comparative study of molecular and ionized gas kinematics in nearby galaxies. These results are based on observations from the EDGE survey, which measured spatially resolved 12CO(J=1-0) in 126 nearby galaxies. Every galaxy in EDGE has corresponding resolved ionized gas measurements from CALIFA. Using a sub-sample of 17 rotation dominated, star-forming galaxies where precise molecular gas rotation curves could be extracted, we derive CO and Hα rotation curves using the same geometric parameters out to ≳1 Re. We find that ~75% of our sample galaxies have smaller ionized gas rotation velocities than the molecular gas in the outer part of the rotation curve. In no case is the molecular gas rotation velocity measurably lower than that of the ionized gas. We suggest that the lower rotation velocity of the ionized gas can be attributed to a significant contribution from extraplanar diffuse ionized gas in a thick, turbulent disk. Observed line ratios tracing diffuse ionized gas are elevated compared to typical values in the midplane of the Milky Way. Kinematic simulations show that a thick disk with a vertical rotation velocity gradient can reproduce the observed differences between the CO and Hα rotation velocities (Levy et al. 2018). Using a sub-sample of edge-on disk galaxies in the EDGE-CALIFA survey, we report preliminary measurements of ionized gas scale heights and vertical rotation velocity gradients. In galaxies affected by this phenomenon, dynamical masses measured using ionized gas rotation curves will be systematically underestimated. This effect will be especially important at high redshifts. Further study of the connection among the kinematics of various ISM phases, stellar feedback, and galaxy dynamics requires high spatial and spectral resolution observations over a range of galaxy properties and redshifts. This field is ready for study with ALMA, the VLA, and the next generation of optical/infrared telescopes and IFUs (e.g. MUSE, SAMI, MIRI).