Step 1: Recall the omega facts. A complex cube root of unity $\omega$ has two key properties: $1 + \omega + \omega^2 = 0$ and $\omega^3 = 1$. From the first, $1 + \omega^2 = -\omega$ and $1 + \omega = -\omega^2$.
Step 2: Simplify the first bracket. Take $1 - \omega + \omega^2$. Group as $(1 + \omega^2) - \omega = -\omega - \omega = -2\omega$.
Step 3: Raise it to the 5th power. Compute $(-2\omega)^5 = -32\omega^5$. Since $\omega^5 = \omega^3\cdot\omega^2 = \omega^2$, this is $-32\omega^2$.
Step 4: Simplify the second bracket. Take $1 + \omega - \omega^2$. Group as $(1 + \omega) - \omega^2 = -\omega^2 - \omega^2 = -2\omega^2$.
Step 5: Raise it to the 5th power. Compute $(-2\omega^2)^5 = -32\omega^{10}$. Since $\omega^{10} = (\omega^3)^3\cdot\omega = \omega$, this is $-32\omega$.
Step 6: Add the two results. Sum them: $-32\omega^2 - 32\omega = -32(\omega + \omega^2)$. Since $\omega + \omega^2 = -1$, this is $-32(-1) = 32$.
\[ \boxed{32} \]