How "big" a difference is pretty much subjective. It depends on what you're doing, what games you're playing, etc.
Games that are very CPU intensive will see more benefit from a motherboard that allows the memory to run at 3200mhz than games that are mainly GPU dependent. Z270 boards are probably going to be easier to find since they are more recent than Z170, but both of them are starting to get pretty scarce.
As far as what you "can play", you "can play" anything, whether at 2133mhz or at 3200mhz, but it's obviously going to offer different results and just how different will depend on the game. As to warzone, who knows. Some users are claiming 100% CPU usage nearly all the time while others say they barely have any. It certainly won't hurt to have dual channel and 8 more GB of RAM than you did before, and until you DO get a new motherboard be sure to go into the BIOS and set the memory speed to 2400mhz if it doesn't do so automatically. Setting the XMP profile won't work because the profile for those sticks won't be supported, but you can certainly manually set the memory speed to 2400mhz after they are installed.