Let's start with stuffing because it's simple -- it means pretty much the same thing in BrE and AmE--a mixture of something bready and some flavo(u)rful things, stuffed into and cooked in another food, especially poultry.
Now, dressing is also general English in that it means generally what 'dresses' a food. Here's what the OED has to say:
So, from this sense we get salad dressing (a general term used in both countries...but the specifics probably deserve their own post. In the meantime, some of it is covered in this old post and especially its comments). But in AmE (and according to mostlyharmless, CanE), dressing can be used specifically to mean 'stuffing'. Many of us have both stuffing and dressing in our vocabularies, which belies the claim that some dialects say one and some the other. The Corpus of Contemporary American English has nine instances of turkey stuffing and three of turkey dressing, all from national publications.
The seasoning substance used in cooking; stuffing; the sauce, etc., used in preparing a dish, a salad, etc.
Some people make a distinction between stuffing and dressing, with stuffing being what is stuffed into the bird (or whatever) and dressing being the same material, but cooked separately. I've been known to make that distinction myself, but I note that the most famous US for a non-stuffed version of this foodstuff is called Stove Top Stuffing. And there are plenty of (North American) people who stuff dressing into turkeys--I suspect that the stuffed-stuffing/non-stuffed-dressing distinction has come about because people found themselves with two words for the same thing and had the natural desire to find a distinction. As Alan Cruse once wrote, "natural languages abhor absolute synonyms just as nature abhors a vacuum".