I'd say 'aufgeben' is something you do with a letter or parcel, something physical, tangible. It also describes the actual act of taking it to the post office or even handing it to the person behind the counter - I wouldn't use 'aufgeben' when I'm just thinking about sending something, like 'Ach, ich muss nächste Woche noch das Paket verschicken'. But 'Ich gehe gleich zur Post um das Paket aufzugeben' is fine, if a little formal.
absenden/abschicken for me are totally synonymous and typical for e-mails. They put the focus on the moment when I hit 'send' rather than describe the general idea of sending something. They are usually used when I'd use 'send off' in English, I'd say.
'verschicken' is what you do with a letter or a parcel or if you want to ship something.
Those are my first thoughts, anyway. Other people might use the words a bit differently, or can think of a better explanation.