I think a good illustration of the difference between "story of/story with" is the title of the chapter in Joseph the Dreamer by K and L Perotta:
What's the Story with Joseph's Dreams?
For those not familiar with the Genesis accounts, Joseph had the power to interpret dreams, his own and others. For example, as a boy, Joseph dreams that the sun and moon and eleven stars bowed down to him; In prison in Egypt, Joseph dreams of seven lean cows eating seven fat ones. What I've just told you are the stories of Joseph's dreams, i.e., the narration of their content. The book chapter on the other hand, goes beyond that giving you not only the narration, but the events that surround them, their interpretations, and the consequences of those interpretations.
Story with is the detailed explanation in context; story of is a simple narration.
In the OP's example, when the reporter asks for the story of the baby's death, he's asking for the timeline -- admission, treatment, crisis, death. What the reporter probably wants is the story with the baby's death -- parental involvement, hospital negligence, and whatever else makes for good headlines.