The FA bans them during takeoff, landing, and whenever the seat belt light is turned on, because in the event of a sudden decelleration, the baby would end up acting as a de facto airbag for the adult.
Most airlines go the FAA one better, and will have FA's insist that you take off the carrier and place it in the overhead bin if you are still wearing it when they do the pre-flight check. (The logic for this being that if there is turbulence and the FA's are told to stay seated, they will not be able to come and make sure that you have indeed taken the baby out of the carrier.)
They are very useful for boarding and debarkation, but you'll want to take it off and store it once you reach your seat.
The issue of FAA approval stickers is for carseats, not soft carriers. In that case, the sticker reads: "This restraint is certified for use in motor vehicles and aircraft." (It does not mention the FAA by name.)