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.

When some little bitchboy is acting like a little tiny pee pee poo poo baby and messing up the group plans.