If $a$ and $b$ are vector, then thinking simply vector 2 norm, $\Vert a \cdot b\Vert = \Vert b\Vert \Vert a\Vert \cos(a,b) $, we know the difference is simply a ratio between the angle of $a$ and $b$.
More generally, in a Hilbert space, Cauchy inequality holds so $$|\langle a,b\rangle|^2 \le \langle a,a\rangle\langle b,b\rangle$$ and we know the only when a, b are parallel, the equality is achieved.
Given two square matrix $A$ and $B$,
when does this happen? $$\Vert AB \Vert = \Vert A \Vert \Vert B \Vert$$
Let's simply assume matrix 2-norm, so $\Vert \cdot \Vert = \Vert \cdot \Vert_2$.