What's the difference between a NM clamp connector and a FMC clamp connector?
A cable clamp has one job

It's supposed to clamp the cable in a way that the cable won't fall out of the hole, and will not damage the cable.  

Notice the clamps are different shapes (assuming these are even pictures of the actual product, who's to say?)

So it goes without saying that the different shape will result in different clamping and damage characteristics. 

However, it looks like the clamps may differ only in the cover plate, so if you have already installed the wrong thing, you may be able to correct it by swapping cover plates instead of rewiring.