I think, generally speaking, longbows tend to be of narrow width with a deeper "core" ....a thicker limb. This style works better with a long working limb, hence, "longbow". Typically, and this is just my opinion, true longbows are 64 plus inches long. They may not be as fast as their shorter counterparts, but they can cast heavier arrows more effectivly, I think, and with little loss of speed. In other words, you can shoot a variety of arrows, and the speed factor will change little. In hunting situations, longbows shine with heavy arrows, and are forgiving and smooth to shoot. That's why we like them.
Flatbows are shorter versions and many incorporate reflex/deflex or vicaversa designs to add speed or forgiveness and they can do this with their thinner limb. I do think there is a trade-off in accuracy. This is more a function of the shooter...I struggle sometimes with a sloppy release, and the longer bows help with that. Also, flatbows will lose cast quicker, the heavier the arrow settup, again, part of the thinner limb design, I think. But the do tend to be faster with lighter weight arrows, and are a joy in heavy brush.