Using Van Allen Probe observations of the inner magnetosphere during geomagnetic storms driven by interplanetary coronal mass ejections (ICMEs) and corotating interaction regions (CIRs), we characterize the impact of these drivers on the storm‐time ring current development. Using 25 ICME‐ and 35 CIR‐driven storms, we have determined the ring current pressure development during the prestorm, main, early‐recovery, and late‐recovery storm phases, as a function of magnetic local time, L shell and ion species (H+, He+, and O+) over the 100‐ to 600‐keV energy range. Consistent with previous results, we find that during the storm main phase, most of the ring current pressure in the inner magnetosphere is contributed by particles on open drift paths drifting duskward leading to a strong partial ring current. The largest difference between the ICME and CIR ring current responses during the storm main and early‐recovery phases is the difference in the response of the <~55‐keV O+ to these drivers. While the H+ pressure response shows similar source and convection patterns for ICME and CIR storms, the O+ pressure response is significantly stronger for ICME storms. The ICME O+ pressure increases more strongly than H+ with decreasing L and peaks at lower L shells than H+.