Hi there Amiguy,
'Aufgeben' usually means to 'give up' ; think of Kate Bush/Peter Gabriel singing, 'Don't Give Up'. Definitely there's a negative connotation there, especially if you're an American :)
When you go to the airport, you don't want to "give up" your passport permanently, and you are not going to "give up" your luggage (for more than a half a day or so).
If you were going to give up your citizenship (that is permanent), that would definitely be 'aufgeben' and not 'abgeben'
Abgeben is more like, 'give over' or 'give to' or 'hand over' - even something abstract like a job or post as someone mentioned here. And there's nothing necessarily 'bad' about it; you are 'giving over' your passport/luggage/responsibility to someone else (perhaps with the expectation or possibility that the thing or job or whatever could be given back to you).
Also, when you give up your citizenship, you're not literally giving (or handing over) YOUR citizenship TO anyone.
In this thread "cookie crumbler" (native German speaker) says you can 'aufgeben' your luggage temporarily by handing over ('abgeben') your suitcase and I suppose that's right, but I still wouldn't want to 'aufgeben' my luggage. That sounds to me like you are leaving it behind because you truly don't want it anymore. At the Frankfurt airport, they have these huge auctions for luggage 'left behind' and not picked up, incredibly, there's enough of that circulating to make that worth it!!!
Obviously the context is important, though, and it's worth a clarification from anyone in a contractual relationship.