There are differences but choosing a pizza stone or a pizza pan for your pizzas is largely a matter of preference.
With a stone, you preheat the oven, build the pizza on the counter, and transfer the pizza from the counter to the hot stone with pizza peel. It requires that you turn the oven on sooner to make certain that the pizza stone is completely heated. (In our kitchen, we turn the oven on a half-hour before starting to make the pizzas.)
With a pizza pan, you build the pizza in the pan and place the pan in the oven. For me, a pan is a little more convenient: I don’t have to remember to turn the oven on early and I don’t have to use a pizza peel. But the difference is minimal.
When you are using stone, your pizza shapes will be freeform. You can make them any size or shape as long as the pizza fits on the stone. You can buy square and round stones. With a pan, the pan determines the size and shape.
It’s important that the crust cooks quickly and completely to avoid soggy, soft pizza crusts. This is especially important with thick crusted or fully loaded pizzas. Lots of meats and cheeses insulate the crust from heat above making bottom heat especially important. With a stone, the crust begins cooking immediately as the dough is placed on the hot stone. There is delay with a pan as the pan heats though but that delay is minimized with a perforated pizza pan*.
Pans are easy to clean and store. Stones are brushed or wiped clean and put away. If you drop a stone, it will break.
In summer, with a stone, you’ll introduce a little more heat in the kitchen since the oven is on longer and the stone takes a while to cool.
Debbie prefers stones. I, Dennis, prefer a perforated pizza pan. But that’s mostly a matter of habit.
*We strongly prefer perforated pans. For thin curst pizzas, it helps them back faster. For thicker crusted pizzas, it helps avoid soggy crusts.