This is a difficult question. I feel a slight difference in meaning here. I will try to explain but I'm not sure I will do it well enough.
give up is not to difficult to understand it has several closely related meanings - stop, surrender, etc.
"give up on something" is more specific in some way for me it has the idea of "give up hope of success"
So if you use "give up the case" it would mean stop investigating it completely for some reason. On the other hand if you use "give up on the case" it means you "give up hope of ever being successful in solving it" this doesn't necessarily mean you have "given up" the case - murder cases are never technically "given up" as in this case new techniques could come along and it could be solved.
I hope that helps but it's difficult to be clear about the difference.
Student or Learner