I have some meaningful experience with every longbow/flatbow configuration on God's green earth. A flatbow by todays standards is typically a shorter (62" or less)bow, with wider limbs than a longer, longbow :-). At those lengths they tend to naturally have wider limbs and shorter riser sections, but there are exceptions. Variety of design in traditional archery is infinite!
One advantage of this design is this; with more mass in the limbs compared to the overall length of many longer longbows (fun with words), it is a very stable bow to shoot. This would be true for a selfbow or glass laminate flatbow. I cannot think of a single disadvantage of the contemporary flatbow.
The Great Northern Bushbow is by far the finest glass laminate flatbow in production. It is the bow of choice for the two bowyers at GN, out of their line of two recurves, one Hill style and two narrow limbed R/D longbows. They have really evolved that design over many years into a true classic: simple, uncluttered, smooth and forgiving, plus you get the advantage of their many years of bow building, representing about a bazillion bows out there.
For a selfbow flatbow go to John Strunk of Oregon. He does not have a website but I can give you his contact info if interested. He is a master selfbowyer who crafts flatbows out of osage, vinemaple, hickory, yew, hazelnut and other woods.