The difference is in the aircraft. Some types of planes have a separate smaller compartment under the nose where these items go, and others do not. If they don't, they go into the main hold, but they go on top of all the other cargo and near the door so that they come off first.
What I use on my stroller is a ripstop sleeve, and over the years it's gotten a lot of black grease on it that it has picked up en route. It protects against rain, though, and going to Florida in summer, that's important, IMO. The wheels and handles stick out of it, so I can pull it behind me down the jetway.
I've always made it a point to use a backpack diaper bag, and NO purse (my wallet is in the pocket of the diaper bag, and my license is in my pocket.) I put the baby down for a few minutes in the gate area, fold the stroller and cover it, and then when boarding is called, I carry the baby in one arm using a sling, and pull the stroller down the jetway with the other hand.
If I have a carseat and I'm alone, I throw myself on the mercy of the gate agent to please try to find an employee who will take the carseat on board and set it down in a window seat for me (not strap it in, they are normally not allowed to do that.)
A fabric sling makes transporting a baby through the airport a lot easier.