Movs clamp voltage, but they have a very soft knee. With rising surge current the clamping voltage will increase. As a typical example, a 100v nominal mov will start conducting 1ma at maybe 80 or 90 volts. At the peak current it maybe nearly 200v clamping, though this could be for currents up to several kA.
Movs are rated to different current levels because they must dissipate the energy as heat. A larger mov at the same voltage rating will take more energy before it is destroyed.
As always, data sheets are your friend and should show you curves for v and I. Most likely this will be the difference for this series.
Edit in response to the follow up comment below:
Choosing an actual mov is not about the down-stream load, but about the level of protection you need to meet. Depending on your industry and country there will be a spec defining the sort of transients which may occur, allowing you to size your mov accordingly.
For a normally operating power delivery system, a mov should have no effect on the equipment at all. It's when problems happen (eg. lightning strike) that your kit needs to survive.
If you are talking about Black box protectors then that is a little different. I would guess that a surge protection plug socket would carry both live and neutral current, and therefore must be rated for a normal operation current. Just like an extension cord, the current carrying copper must be able to supply the load. This however has nothing to do with the mov rating.