Quarry tile- low conductivity, low thermal mass, inexpensive, can be hard to find, certain brands can be especially fragile, can have low resistance to thermal shock (grill use can be iffy).
Cordierite- better conductivity, good thermal mass (if purchased thick enough) less expensive than fibrament, easy to order online, sturdy (usually), high resistance to thermal shock (grill friendly).
Fibrament- low conductivity, expensive, easy to order, sturdier than quarry tile, but less sturdy than cordierite, poor resistance to thermal shock (only works in a grill with a metal deflector).
The most important feature of all these materials is conductivity. The conductivity dictates the transfer of heat, which in turn, dictates the bake time. With Chicago thin crusts, you can pretty much use any material you want because the bake time is so slow, but as you get into NY style, you need fast bake times, and, in ovens with broilers, you need conductive materials- ideally more conductive than cordierite. If the oven has the broiler in a separate drawer, then, for NY, you'll need a less conductive material, such as quarry tile along with a higher, modded oven temp.
Long story short, there is no 'one size fits all best stone,' only the best stone(s) for a particular scenario.
Quarry tile can work well for bread, although I'd seek out a tile that's a bit on the dense/solid side rather than one that feels a bit light for it's size. The denser the tile, the less air, the less air, the greater the conductivity. Quarry tiles might also benefit, on the bread side, from stacking to increase the thermal mass a bit.
