What gives density matrix the expressive power to be able to represent mixture of pure states?

For example if $|\Omega\rangle$ is 50-50 mixture of $|\Psi\rangle=\frac{1}{\sqrt{2}}(|u\rangle+|d\rangle)$ and $|\Phi\rangle=\frac{1}{\sqrt{2}}(|u\rangle-|d\rangle)$ we can not represent such as $|\Omega\rangle=\frac{1}{2}(|\Psi\rangle+|\Phi\rangle)$, but we can represent it as $\rho=\frac{1}{2}(|\Psi\rangle\langle\Psi|+|\Phi\rangle\langle\Phi|)$.

Intuitively what makes $|\Psi\rangle\langle\Psi|$ able to represent such as opposed to $|\Psi\rangle$?