Not giving a shit has to do with the level of care given. Not to be confused with "not giving a fuck" see that definition. Not giving a shit means that you literally don't care enough to take action.
For example, Dillon may not want to go to work today. If he doesn't give a fuck, he will still go, but be filled with apathy. If he doesn't give a shit, then he won't bother showing up.
The kitchen staff are not giving a shit anymore so they don't do their fuckingcleaning duties.
Simon: Hey, what does not giving a shit about the Third World mean?
Justin: Basically it means you don't care about the problems of people from the most poor nations, duh.
Ruler of Lake Titicaca. Rumored to have a bunghole that gets very angry if it does not receive toilet paper. Cornholio the Great is often seen walking around with his shirt over his head and his hands in the air, chanting songs about his power, and his bunghole.
"I am Cornholio! You do not want to face the wrath of my bunghole, for I need TP!"
Butthead: Shut up, Beavis! (uh huh huh huh)
Beavis: Um, okay. (heh heh heh heh).