Each frame of a 60Hz game spans 16.66ms, while in a 30Hz game it's 33.33ms. Depending on how well the engine is coded, there will be at least one frame of latency between input and reactions, and maybe more. So we could be looking at the difference between 33 and 66 milliseconds of input latency, at least. This doesn't even consider latency from input to the computer, or from the computer to the display.
Now consider that we react to events within about 200-300ms. 66ms of latency on top is going slow things down considerably, and when games are poorly coded and dump a few additional frames on top - which could happen because of propagation through events and subsystems, network latency, or animation concerns - the control starts to feel "wrong" in a hard-to-identify way.