If your imac has 1333mhz ram installed and it is speced to use that speed ram then it is not a late 2012 model. Or if you upgraded the ram when you bought it from apple they installed the wrong speed ram. Or if you upgraded the ram yourself after you bought it you bought and installed the wrong ram. Late 2012 imacs use 1600mhz ram.
Anyway I would not install 1333mhz ram in a system that specs 1600mhz ram. So I would not switch them. If your imac is speced for 1333mhz and you install 1600mhz it may or may not work and especially if the mini specs 1600mhz and you install 1333mhz I doubt it would work.
Just order the correct ram for the mini. Ram is cheap these days.
A gain of 8GB for $125 kind of *****. However, your point has been well taken. As it happens, I did upgrade the RAM in the iMac prior and apparently missed the Mhz rating. Interestingly, the number I gave you for the 1333Mhz came from within the system profiler on the iMac and states "Your Mac contains 4 memory slots, each of which accepts a 1333Mhz DDR3 memory module." I figured that it would show me what it is designed to use here since the actual RAM is listed in the over tab. Apparently this reflects what is installed and not what the system was designed for. I cross checked what you said and of course, you're correct in that both Macs run the same RAM...which now means I get to decrease my iMac RAM and still spend too much money on an 8GB increase.