I'm not sure I understand your question, because it title hints that you know the answer, or at least you know more than you think you do.
Anyway, when you go up the stairs your movement has a horizontal component (forward) and a vertical component (up). The horizontal component is not affected by gravity, but the vertical component is. When you go down the stairs, the vertical component changes from up to down, or, alternatively, from going "against" gravity to going "with" it. That's why the difference gets bigger when the stairs are steeper.
Did I answer your question? I feel like I missed something there...