They are both halogenated compounds. The pool chemical stuff is just made convenient and includes "stabilizer" (Cyanuric acid); that's important for sunlight-hit pools but of no consequence in this case.
In THIS use, there is no material difference between bromine and chlorine. In pool use there is quite a bit of difference, the biggest being that once you use bromine you're stuck with it unless you COMPLETELY drain the pool and refill it!
The potential problem with these chemicals is that they can evolve more agent than can be absorbed in the water, which leads to free gas in the system when its shut down. Free chlorine or bromine gas is extremely corrosive to almost all metals.
It requires about 8ppm to "inactivate" biological processes, and over long periods of time that is enough to kill biological things in the water. This sort of thing works because the water is basically poisoned enough to keep things from being able to get growing in the pipes. This is a common and established way to prevent biological fouling of piping in various industrial processes that use raw water, and properly controlled it works with little or no risk to the piping or the environment after discharge.
The issue using it this way is the lack of control. If you get free gas (chlorine or bromine) in the system after shutdown and that gas ends up in a metal pipe, you can expect severe damage. I have seen copper feed pipe into a pool filter completely eaten through within a couple of years by improper tablet feeder installation!
If you're going to use it pay very, very close attention to your strainers and baskets (where you're going to put the tablets) and I would also, if possible, put a riser loop in the piping to prevent any evolved bubble of gas from ending up in places you really don't want it (like in your A/C condensors!) Basdically, the loop traps the gas bubble in the hose - while it will degrade the hose pretty badly too, hose is a lot cheaper than an AC condensing unit!
Expect to change that riser loop of hose out every year.....