Quoted By:
I would say they're a special operations force that combines environmental protection, disaster response, and search and rescue. So... park rangers, basically.
If we want to get super autistic over what gives them jurisdiction to act basically anywhere, I would suppose either places where they appear are either regional parks and this is not made clear to the player, or they are empowered to patrol the entire sovereign territory of whatever state they belong to. My guess would be the latter.
As to why some rangers use capture stylers and others use Poke Balls, it has to be a cultural thing. It might also be backed by legal, who knows, but if it was a strictly legal limitation, surely the criminal organizations wouldn't be so opposed to having that tool at their disposal in regions where they are banned. What's one more law broken while we're already breaking all the other ones, huh? But if it was a cultural thing embedded in the local society, I'd imagine the criminals may turn their noses up against the idea of using Poke Balls in places where they are culturally frowned upon, preferring other means to achieve their ends.