Consider two rectangles with the same length $\ell$ but different widths. Let the widths be $w_1$ and $w_2$ where $w_1 - w_2 = 6$.

The perimeters are:

$P_1 = 2\ell + 2w_1$ $P_2 = 2\ell + 2w_2$

The difference in perimeters:

$P_1 - P_2 = 2\ell + 2w_1 - 2\ell - 2w_2 = 2(w_1 - w_2) = 2(6) = 12$

The lengths cancel out, so the perimeter difference is $12$.