If that wrap is real metal (try a magnet) and not plastic, what you are seeing is simple corrosion. If the wrap is metal, you can use #0000 steel wool to very lightly burnish it away. (I know, I know - several people are pulling out their hair right now yelling, "Don't use steel wool!" But if you use caution and don't get heavy-handed, you can remove the corrosion quite easily without scratching the shell significantly. At this point, what do you have to lose? You are fighting rust, after all. Not to mention that using Coke and a wad of aluminum foil somehow makes me squirm like nails on a blackboard. Eeeeeeeeeek!) Go ahead and disassemble the whole snare and clean everything else while you're at it.
To stop the rust from returning (believe me, I've tried this and it works), simply dip your finger in some heavy axle grease and rub it vigorously into the rust spots, then wipe off the excess with a cotton cloth. Don't work too hard at this. The idea is to clean off the excess grease while leaving enough behind - just a film - to stop the rust. And it will not come off on your clothes.
Alternatively, there are a number of chemical rust treatments that will stop it dead in its tracks, but these compounds usually blacken the rusted area.
It's likely that subsequent posters may have even better ideas than mine, but this is how I've done it with success in the past. I'd love to hear of alternatives, too.
GeeDeeEmm