Outside skirt all the way!!!!! Inside skirt is thinner and much tougher, even when sliced properly against the grain.
Outside Skirt is kind of a pain to clean as there is a membrane on either side of the meat, but if you have a sharp boning knife you can peel/tear it off pretty easily, just make sure you are not tearing the meat underneath.
As others have mentioned, you almost never find outside skirt in grocery stores as most of it gets exported or sold to restaurants. It's thicker than inside skirt and it loves to be charred over super-high heat keeping the inside medium rare. Simple marinades like Chimichurri sauce or just salt and pepper work best. Many Brazilian Steakhouses feature Outside Skirt for "Churasco".
To me it has an excellent "beefy" taste unlike anything else on the cow.
