Bridgestone bought Firestone several years ago when Firestone value dropped because of a huge problem with original equipment tires on Ford Explorers. So Firestone and Bridgestone are both made by the same company in the same plants. Tires with either name can be identical, although Bridgestone specializes in truck tires while Firestone leans more towards car tires. They both make LT and P-series tires. So the exact tread would be the only difference between a Firestone and a Bridgestone of the same construction.
I have P-series Dueler H/L Alenzas on my pickup, and they are fine. Not cheap at over $200 each. If you can find a store that carries both brands, then observe the tread differences, and you may find no difference in some but huge differences in others. My Duelers are A/S (all season) tread, but they also make several different A/S treads. According to the catalog, even the Dueler A/Ts have all-season treads, so look critically at each different Dueler name to decide exactly which one you want. The Dueler website might give you some info, but the only real test is several thousand miles under different real-world conditions.