BEB stands for brass enclosed base. The bullet has a brass jacket around it that is supposed to reduce lead fouling. I have shot some of it in 9mm, and have a box of .380 that I haven't tried yet. The 9mm works just fine in my Taurus PT111, and I really didn't notice any difference in the fouling in the barrel. I clean my pistols every time I shoot so I would only see a difference if I shot a lot at one time. Personally, I think it's a ad gimmick, but doesn't cost any more than regular white box. Haven't seen any of it in my local Wally World lately, maybe it didn't sell well.
When I got my first BT380. it did not like the Winchester blunt nose rounds. Had a very high rate of FTC. Found out later that is not uncommon in a new BT380. That was several years ago, and I've not tried them again.