There is very little difference between these flies and, quite frankly, you could use both flies to imitate the other one.
Generally speaking, an Adams will have a gray fur body and wings made of grizzly hackle tips whereas a Henderickson will usually have a body made of wrapped quill and wings of mallard flank or similar material. Honestly, I'd regard 'em as essentially the same fly with no real difference in the eye of the trout (no difference due to the materials that is. Size can make considerable difference).