Easy answer.
AutoX focuses primarily on low speed turns and agility. A wider tire facilitates faster turning.
The faster turning from a wider tires also translates onto a track. HOWEVER, the wider tire also introduces more rolling resistance as well as more rotational inertia. At every road course we've been on, the additional power put to the ground from having a skinnier tire has been more than worth the loss in raw cornering speed.
In autoX, you don't spend nearly as much time accelerating as you do on a track.
We started with skinny tires, and worked our way up, and found 225 to generally be fastest for us. Robispec also started with wider tires, and went even wider, before going narrower.
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Austin: You have to tiptoe around the rear end and putting down power on a s2k (more so on my s2k than yours), which is why the S2k benefits from a wider tire. It also has a fatter powerband, and SIGNIFICANTLY more aggressive gearing.
For everyone else: 4th gear in the BRZ has the approximately the same mechanical torque output as 6th gear on the s2k (6th on the s2k is slightly less)