The two flours do not have the same absorption rate (hydration). The absorption rate of all-purpose flour is about 5-7% higher than 00 flour--such as the Caputo Pizzeria flour. However, I have found that if I add the flour gradually to the water and let it get absorbed gradually, I am able to get to similar hydration rates out of both flours. I weigh the water and flour, but that is not the reason why your results have been inconsistent. Assuming your volume measurements are fairly consistent from batch to batch, which may or not be true, it's mainly the differences in the two flours and how you hydrate them.
What you may want to do with the 00 flour is to add the amount of water you want to use (based on your recipe) to the mixer bowl (usually with the salt dissolved in it), and gradually add the flour (or flour/yeast mixture) while your mixer is at Stir speed. When I do it, I use a thin-blade spatula to help move the flour into the path of the dough hook to get everything mixed together as best as possible. I keep adding the flour a couple of tablespoons at a time, also at Stir speed, until I get the desired dough consistency and feel. When using oil, I usually add it toward the end of the knead process, and continue kneading at Stir or 1 speed. You want the final dough to be smooth and soft and a bit on the damp (tacky) side.
I suspect what happened in your case with the 00 dough is that the hydration was too high and the bake time was too long, causing the moisture in the dough (which is usually thin to begin with) to be driven out of the dough, producing a cracker-like crust by the time the crust has started to develop some color. This is a problem inherent in using a 00 dough in a home oven, for which it was not intended. To get better results, you may want to use a thicker dough to begin with and, if you are not already doing so, add some oil to the dough.