Headboards on Bermudian sails have been getting fatter and fatter as the years have progressed. With the advent of full battens, headboards have dramatically increased in size, so much so that they can't use the typical headboard as it would be too heavy. Now you commonly see a small, traditional headboard to which the halyard attaches and a diagonal batten to hold up the extended leach. These are still Bermudians, even though the line is visually getting more blurry.
The primary differences are the gaff is a spar, while the bat wing, square headed Bermudians typically don't have a spar, but a headboard and/or batten. Even though a car batten could have length adjustments in it, I wouldn't call this an outhaul tackle like a spar would have, nor is the sail attached to the spar like a gaffer, as the batten is inserted inside the sail. There are other differences, particularly in shape and rigging, but primarily the gaff rig has a spar at the head of the sail, while a Bermudian has localized stiffeners inserted into the sail and/or a headboard that may serve similar purposes.