Simple question is, "Where did the energy go?" First guess would be that the bodies of the fat eaters produced more heat, perhaps on account of a less efficient metabolism of fat. Couldn't that be measured directly with a thermometer? Other guess would be that the fat eaters were more active, producing more mechanical energy. That would be a very interesting possibility to explore. A few decades back, my reading led me to think that conventional wisdom was that fat-burning exercise was aerobic, ie requiring oxygen and breathing. Could burning fat require more exertion to acquire the oxygen?
Does the “not all calories are identical” assertion reduce to "the number of calories of heat obtained when a food is burned in a flame outside the body (the number of calories on the label) is not equal to the net number of useful calories obtained when the food is metabolized?"
250 calories per day is only about 1 kg per month of weight loss. This is not trivial compared to typical weight loss results, but it is quite a bit less than what people searching for diet advice would consider a game-changer. And it is not clear if there are other consequences of changing the composition of the diet over the long term of steady weight loss that most dieters need.