Intuition behind unitary equivalence

Question

We say that two elements $a,b$ in some $C^*$-algebra $\mathcal{A}$ are unitarily equivalent if there exists a unitary $u$ in the unitarization of $\mathcal{A}$ such that $a = ubu^*$. Is it possible to provide some simple intuition regarding how similar $a$ and $b$ are, when this condition holds? And similarly, can one obtain some intuition when $a$ and $b$ are approximately unitarily equivalent?