This is a both simple question and a very complex one: the simple answer is that bran itself, while great, is just a delivery vehicle for the real “functional ingredients” such as fiber.
Bran and fiber are less digestible by human (stomach) enzymes. So they go into the small intestine and ideally the large intestine where they feed the gut microbes, which have all kinds of health benefits… including increasing bone density in humans by increased nutrient (calcium) uptake in the large intestine/colon.
There is no better or best fiber. They are all good- look it up- there are 0 drawbacks (other than gas, but even that is 100% a very very good thing to have happen as it means the fiber is working). Mixing up your diet with as many fibers is possible is best. But any individual is great!
If you’re asking, of those two ingredients, which is better for your health, I think the answer is that they are different. Whole wheat includes a small percentage of bran, as well as the wheat germ (a lot of nutrients, healthy fat, and also some fiber) and the endosperm (which is the part that makes gluten and bread). The bran is added to increase the bran benefits, of which there is a relatively small amount in whole wheat by itself.
Here is a diagram of the different parts of a whole wheat kernel. If it’s not clear from the drawing, the bran is the outer coating.
I eat generic bran flakes dry. I get around 7g of fiber in 1 cup. I think oats are 4 or 5 per half cup serving; I eat generic quick oats with hot water and blackberries (which I believe have the highest fiber for a fruit or perhaps just highest fiber to calorie ratio). I try to add a daily serving of leafy green vegetables, also. So I get enough fiber for almost no money and very few of my daily calories. I have to make sure I drink a lot of water, though.
Anyone ever make Dukan Pancakes? Beat an egg, add 2 Tb Greek yogurt and 2 Tb oat bran. Add some cinnamon if you like. This makes a thick batter that you cook like a pancake. Delicious.
In general the “functional ingredients” matter and so does the delivery vehicle, the package, even if for fiber the benefit of adequate fiber of various sorts is very large. Using bran as one way to increase fiber intake is great and results in major benefits; aiming for adequate fiber intake by choosing more whole foods that are high in fiber is even better. For many foods the packaging that is associated with having high fiber results in greater satiety, leads to release at different points in the digestive tract, and is a marker for high probability of being copackaged with a wide variety of health promoting components. Crap supplemented with fiber is not as good for us as health whole food with slightly less fiber.