I've been using bleach in my tub for over a year now and I can now give my opinion on chlorine vs bromine.
Chlorine is more familiar to use (for me) and cheaper than bromine. You need to stay up on it though, basically adding a small amount of bleach every other day.
Chloromines have turned out to be a problem, "That chlorine smell" of the chloromines gets trapped under the cover kinda like an indoor pool but much worse. In fact it's really jazzed up the underneath of the cover.
I've switched to a 1" bromine tablet float and occasionally add bleach to the tub now. The floater/feeder keeps the level more consistent and I don't have to worry about the chloromines smell. Bromine does have it's own smell, so regardless the tub water will always have an odor when compared to fresh filled water.