Example 4 Fill the 9 cells in Figure 8 with the numbers $1 \sim 9$ respectively, so that the sum of the differences (the larger number minus the smaller number) of the adjacent (having a common edge) two numbers in these 9 cells is maximized. Find this maximum value.
To maximize the sum of the 9 differences, we hope that the 9 minuends are as large as possible and the 9 subtrahends are as small as possible. For example, the largest number 9 should be adjacent to the smallest number 1 and the second smallest number 2, the second largest number 8 should be adjacent to 1, and then to 3, the third largest number 7 should be adjacent to 2, and then to 4, the fourth largest number 6 should be adjacent to 3, and the remaining 5 should be placed in the cell between 4 and 6 (as shown in Figure 9). Intuitively, this construction maximizes the differences, and their total sum is
\[
\begin{array}{l}
(9-1)+(9-2)+(8-1)+(8-3)+ \\
(7-2)+(7-4)+(6-3)+(6-5)+ \\
(5-4) \\
=(9+8+7+6) \times 2+5-5-(1+2+3+4) \times 2 \\
=40 .
\end{array}
\]

As analyzed in Example 1, this is the maximum value.