Don't panic, you didn't do anything wrong.
Fin rot isn't caused by a specific "bad bacteria" that is present in the one tank and not the other. All of our tanks contain different bacteria, fungal spores etc, it's just that usually they don't present a problem unless there's a weakness in the fish that lets it in you know? You're not introducing some rare pathogen to your 'healthy' tank, it's likely in there already.
Think of it like this... you're doing some gardening, or playing around in the dirt in your backyard. Not a problem usually, since your skin protects you from all the bacteria in the soil. But if you have an open wound, and the dirt gets into that, you're likely to get a bacterial infection.
Fin rot, like a lot of bacterial and fungal infections, is a secondary infection. Usually the fish has been in poor water quality, the fins start to get tiny tears and the fishes immune system is weaker because it's stressed by the water parameters already. Bacteria that were always present, now infect the fins, because there's an opening there and the fishes immune system can't fight off the infection the way a healthier, non-stressed fish in cleaner water might.
So keep the water as immaculate as you can. Medicate the fin rot if the infection is bad enough that clean water and salt treatment are no longer enough. Your healthy tank is fine, as long as you've kept up with maintenance and the fish are healthy.
there is another healthy betta in there and a snail
