Choice A is correct. The area $A$, in square inches, of a rectangle is the product of its length $\mathcal{l}$, in inches, and its width $w$, in inches; thus, $A=\mathcal{l}w$. It's given that the area of a rectangle is $630$ square inches and the length of the rectangle is $70$ inches. Substituting $630$ for $A$ and $70$ for $\mathcal{l}$ in the equation $A=\mathcal{l}w$ yields $630=70w$. Dividing both sides of this equation by $70$ yields $9=w$. Therefore, the width, in inches, of this rectangle is $9$.

Choice B is incorrect. This is the length, not the width, in inches, of the rectangle.

Choice C is incorrect. This is half the area, in square inches, not the width, in inches, of the rectangle.

Choice D is incorrect. This is the difference between the area, in square inches, and the length, in inches, of the rectangle, not the width, in inches, of the rectangle.