I've used both and not found much difference. Both make excellent tires and the price is the only thing that would sway me between them.
Yes, Bridgestone makes Firestone tires. The company is now based in Japan.
The Bridgestone Corporation of Japan makes Bridgestone tires
Falken
I have falken tires and they say made in Thailand .
Falken tires are made in Japan by Sumitomo Ohtsu Rubber company
Every few years I always get Goodyear tires. I always feel safe driving with that brand. I've never had a blowout or anything. Bridgestone, Michelin, and Falken are all highly-rated tire brands.
Ironman Tires are made by Hercules USA, who also make Cooper, Continental and Falken tires.
Primewell tires are manufactured in Asia for sale in North America by Bridgestone/Firestone. This is not the same as "Primewell tires are built by Bridgestone in the same plants as the flagship Bridgestone models".
Bridgestone Tire Company
Falken Azenis tires can be bought on the official website of Falken, where one can use an built-in function to locate a store that sells the tires. Alternatively one can also buy the tires from many other websites that sell tires.
There are multiple companies that manufacture tires with model number P235 70R16. These include: BF Goodrich, Bridgestone, Continental, Cooper, Definity, Falken, Firestone, Futura, and Goodyear, to name a few.