In each important rectangle, the number of black squares is one more than the number of white squares. Hence, each important rectangle contributes $+1$ to the difference $B-W$. The value of $B-W$ is thus the same as the number of important rectangles on the board.

Let us number the rows on the board $1, 2, \ldots, 8$ from the top downwards and the columns $1, 2, \ldots, 8$ from the left to the right. So $(1, 1)$ is the upper left square and $(8, 8)$ denotes the lower right square. Assume $(1, 1)$ is a black square. Then all $(i, j)$ with both $i$ and $j$ odd, as well as all those with both $i$ and $j$ even, are black squares. All other squares are white.

By focusing only on the four odd-numbered rows and the four odd-numbered columns, we find that they determine $(4 + \binom{4}{2})^2 = 100$ important rectangles. Similarly, the four even-numbered rows and the four even-numbered columns determine another $100$ important rectangles, giving a total of $200$ important rectangles on the board. It follows that $B-W = 200$.