I think of exurbs as areas that aren't connected to the city via a solid strip of urban land, but still are heavily economically tied to the city. I think this is more common in the West where there are a lot of reasonably dense urban areas with virtually empty land between them. Suburbs are just parts of an urban area that aren't politically part of the core city of that area for whatever reason (although in my experienced it tends to refer to smaller and often wealthier examples of those communities).
"Outskirts" I hear more in reference to small towns, and it refers to the rural area immediately outside of the town. If someone told me they were from the "outskirts of Boston" I wouldn't really know what they meant.
Interestingly, if someone told me they lived on the outskirts of Boston I would assume a town near I-495 without a second thought. I went to school in the suburbs of Boston (ie near rt 128), but my in-laws are from just inside the outer ring of I-495. I've even heard them refer to their towns as on the outskirts of Boston.
