How are they related?
By the 1998 Vacancies Act. That is the law that allows the President to replace a person in an office that normally requires Senate confirmation with anybody else who was confirmed by the Senate, even if the replacement was confirmed for a job requiring entirely different abilities and knowledge.
The language of the Vacancies Act says that it applies when a Senate-confirmed officer “dies, resigns, or is otherwise unable to perform the functions and duties of the office.” You notice what is missing there? Congress did not say ”resigns, is dismissed, or is otherwise...”
Congress may well have left that out for good reason: enabling the President to shuffle people around without Senate review on his own strategy or whim, rather than when forced to do so by external events, is a broad concession of oversight powers, without the practical need that a death or resignation creates.
Defenders of Presidential authority would argue differently. “Otherwise unable to perform” is broad language, they would say—broad enough to cover a firing. “Is Tillerson able to perform his duties? No—his key doesn’t work any more. The Act applies.”
This is the sort of dispute that is resolved only in court, and it hasn’t gone to court yet.
But what about Sessions?
One factor in Trump’s reluctance to dismiss Sessions is this exact question. If he can replace him immediately with a Pompeo—who would then say “Hey, I’m not recused”—he can end the Mueller investigation. If he cannot use the Vacancies Act in the case of a firing, that doesn’t work: we get an acting AG from the ranks (Rosenstein, I believe).
So: today Trump is (among other things) running a test case and trying to establish a precedent. If Pompeo takes over at State without any challenge, it becomes easier to replace Sessions with a more compliant puppet.