Often, when used to mean "extremely" they can be similar. But this isn't always true. It seems to me the most common usages of the words is with beautiful or expensive. A woman can be staggering beautiful or stunning or stunningly beautiful. An event can be staggeringly expensive but usually wouldn't be stunningly expensive. You could be either staggered or stunned by the cost. You might be stunned by an action, but less likely to be staggered by an action. This is what comes to mind without a dictionary, which we both probably ought to consult to have a good answer here.

I think you can use it any time you want to stress that you are really amazed and surprised by something. Stunning is usually a positive thing- Her beauty was stunning, the view from our hotel room was stunning. But "staggering" is just something that surprises you,not just a positive thing. For example "the statistics about the number of divorces in our society are just staggering!" it means we are surprised and shocked by the number of divorces in society. I hope this helps.