Well, a Birkin is a tote bag. It has two handles. A Kelly has a single top handle, and new ones come with a shoulder strap. The Kelly is NOT a tote bag and is not meant for "stuffing", whereas Birkins can take a lot of abuse via overstuffing, etc. A Kelly is generally considered to be slightly less casual than a Birkin (though either bag in certain leathers can be much more dressy or much more casual).
That's the basic difference though of course all of us here on the H subforum could write a book about the rest of the nuances between the two styles of bag.