Practical differences of two types of tile levelling system
I agree that these leveling systems are both more effective for larger format tiles. However, I think the question is good and one I had trouble finding an answer to when I was tiling my kitchen.

Re-use


Wedge: The wedge is re-usable, the plastic tab is broken off.
Screw/Spin: The spin top is re-usable, the plastic tab is broken off.


Tools


Wedge: To get the right pressure, a special tool is needed
Screw/Spin: The spin top is re-usable, the plastic tab is broken off. No special tools are needed.


Corners


Wedge: The wedges are really intended to go across two tiles, so for a corner, you need 4 wedges.
Screw/Spin: The spin top can sit directly on a corner or a T and provide leveling across each tile.


Shifting


Wedge: The wedge pushes both tiles in the same direction, which is not a problem if you have spacers between the tiles.
Screw/Spin: The spinning action of the top can cause the tiles to shift in different directions as you tighten them. This can cause the tile to twist slightly, even with spacers.


Availability

Both systems were available through a certain orange big box store (I found the wedge-type in-stock, and the Screw type needed to be ordered online). 

My Experience

I ended up trying the wedge type first, then switched to the screw type. I found that the upward pressure of the wedge on the plastic could cause the tab to break prematurely (and you have to pull the tile up to put a new tab in). With the screw type, I could tighten, loosen, re-tighten as needed. I could also put one screw type at the corner intersection of 4 tiles. In the end, I was very happy with the result using the screw-type.