Let $X$, $Y$ be Hilbert spaces. Let $S, T : X \rightarrow Y$ be unbounded operator.
Suppose $S$ and $T$ be bounded operators. Then we can compare by their maximum distance on the unit ball of $X$.
Suppose $S$ and $T$ are unbounded but closed operators, with the same set of definition. Then we can compare them locally on the unit ball, i.e. for every point $x$ on unit ball of $X$ which lies in the domain of $S$ and $T$ there exists a neighbourhood $U(x) \in S^1(X)$ such that the distance of $S$ and $T$ is bounded on the intersection of the domain of $S$ and $T$ with $U(x)$. (Can we?). This means we compare their graphs.
Is there something like this for general unbounded operators? I can only imagine comparing them on finite-dimensional subspaces, to get some sense of distance.