Is there a difference in the end result between using mechanical leaveners vs chemical leaveners in a cake?
"Leavener" as you're defining the term, is vague. A meringue of whipped egg whites are "leaveners" in the sense that they provide a stable structure for tiny air molecules; but of course an egg white by itself won't leaven anything, because there's no gas for the foam. Baking powder is a "leavener" in the sense of generating gas for a foam, but a mixture of baking powder and water won't be leavened, because the water won't support the foam. In a vague sense, meringue is a very potent leavener because it can hold so much gas, but its structure is easily broken up by fats (this is why it loses so much volume when folded into a batter, yet not when folded into itself). A batter based on flour and whole eggs is more durable but will never hold as much gas as a meringue while maintaining a "cakey" texture.

If you replaced egg whites with baking powder in an angel food cake, I suspect you'd get a much denser result, because the (low-gluten) flour and egg yolks would not hold the gas in well. Likewise, a meringue would not work very well in a high-fat cake because its stability would be compromised. The latter might produce an acceptable result, but I can't think why you'd want to do it.