What these two beef cuts share in common is that they come from the underside of the cow, but they both have distinctly different tastes and textures.
Flank is the leanest of the two. It's a good all-purpose beef cut, suitable for grilling, roasting, broiling, or sautéing. But because it's so lean, it can be dry and tough if overcooked or sliced too thickly—it's important to cook flank to no more than medium and slice it very thinly against the grain.
Skirt steak is often used in Mexican fajitas. It's much more coarsely grained than flank and quite a bit fattier as well, but that extra marbling gives the meat a lovely richness when grilled hot and fast over a blazing fire or seared in a cast-iron pan (it's a thinner cut than the other two, so it needs high temps to get a delicious crust outside without overcooking it inside). The skirt cut is long, and the grain runs across its narrow width. It's extremely important to slice thinly against the grain—cut skirt with the grain and it'll be too tough to swallow.