What is $5^\circ \mathrm{F}$ minus $5^\circ \mathrm{F}$? Is it $0^\circ \mathrm{F}$ or is it $0\,\mathrm{K}$ (Kelvin)?
From an arithmetic standpoint, it seems like it should be $0^\circ \mathrm{F}$, but that seems inconsistent because the result represents a delta ($0 \,\Delta^\circ \mathrm{F}$ perhaps) which is conceptually completely different. Doing math with units like $^\circ\mathrm{F}/s$ completely breaks down.
From a natural sciences standpoint, it's 0 K, because an amount of heat minus the same amount of heat is no heat, but that seems inconsistent because everyone knows that $5 - 5 = 0$, and this would instead yield $5^\circ \mathrm{F} - 5^\circ\mathrm{F} = -459.67^\circ\mathrm{F}$, which may be surprising.
Is there a convention for doing arithmetic on Interval Scale measurements? That same page suggests there is a difference operation for them in its comparison table.
For context, I'm working on a Python library for doing dimensioned arithmetic (with an eye towards the natural sciences) and interval scales are making things complicated. For instance, $^\circ\mathrm{F}\cdot s$ doesn't seem to represent a meaningful physical dimension; it can't be converted to $\mathrm{K}\cdot s$. For example, trying to convert $6^\circ\mathrm{C}\cdot s$ by visualizing temperature and time as axes:


You seem to have the correct idea that it's important to distinguish between temperatures and temperature changes. So to answer your question, you'll need to first decide what the question is, i.e., do those two occurrences of $5^\circ F$ in the question refer to temperatures or to temperature changes?
For example, if they're both temperatures, then subtracting produces a temperature change of $0^\circ F$, which is the same as a temperature change of $0\ K$.
For another example, if the first $5^\circ F$ is a temperature and the second is a temperature change, then subtracting produces a temperature of $0^\circ F$, which is very different from a temperature of $0\ K$.
