I really don't understand how anyone can make this distinction. Politicians are expected to talk about their personal life. We want to know how their decisions will be shaped by their experiences, their family and their faith. President Bush won a long battle with alcoholism and I think that telling people about that demonstrated something important about his personal resolve — or was he just "exploiting" his personal victory to "acquire political power"?
If "doing so to acquire political power" just means "in a speech or publication during a campaign" then this "bad character" flaw could be applied to any politician of any stripe during any campaign. The death of his child was a major event in the life of John Edwards and it's perfectly fine for him to explain how it shaped his thinking. You can certainly criticize the thinking.
Are there people cynical enough to use their personal tragedies for personal gain? Sure, but how do you determine that? The answer here seems to be "because they are saying something I don't like." If that's the test people want to use, they are free to do so, but they shouldn't pretend to be standing on high moral ground when they do.
A politician that uses a personal tragedy to argue for a policy change is exploiting the tragedy, because to fight against their policy will be seen as attacking them for their tragedy. “How can you question Gore’s commitment to stopping the scourge of smoking, he lost a sister to cancer!!!”
How often do the do it? Is it brought up in context in which it is not appropriate? Do they reach out to political opponents who share the same tragedies? Do they use it to insulate themselves from attacks on the policies they are advocating?
The answer here seems to be "because they are saying something I don't like."
Concerning the Edwards and the Jersey Girls, it's because they are appear to be using tragedy in the pursuit of power and wealth.