I figure, that with the same amount of food, since it could be cooked in either, especially if one only has the Dutch oven, there might be more radiant heat hitting the surface of the food. Less of that radiant heat turning into convective heat. I don't know if that could make a difference.Both the French oven and the brasier are closed vessels. When used in the oven for braising, I'd choose the one that was the most appropriate size. Brasier for short ribs, French oven for a pot roast, pot of stew or chili. I don't see much of a material difference other than size.
OTOH, just because those cooking vessels come with lids doesn't mean that one will automatically decide to use the lid. Okay, most likely with the lid on, but not guaranteed.