What are the difference between outside and inside skirt steak?
Per Kenji Alt's Food Lab article on fajitas:


  [T]here are really two distinct cuts from the diaphragm of the steer.
  
  The outside skirt steak is from the plate section, below the rib and
  between the brisket and flank, and usually comes with the membrane
  still attached, which needs to be trimmed before cooking.
  
  Inside skirt comes from the flank—it's narrower and thinner than the
  outside skirt, and comes with the membrane removed.


According to the Houston Press, outside skirt stake is extremely rare in the US, as much of it gets exported to Japan due to a quick in the tariffs.  The outside cut is tougher, but extremely flavorful.  Both are used in similar applications, if you can get them.

Given that any skirt you buy is probably inside skirt, the difference is likely to be moot.  In making fajitas, Kenji Alt recommends grilling it over extremely high heat, to get browning flavors on the outside, and not overcook the center.  He recommends not cooking it over medium rare.

A couple of pictures: