There has definitely been a lot of confusion and debate over whether Tarkin has authority over Vader or not (myself included). I think the (canon) books and comics clear up this issue to a very good degree. I'll outline the facts, as well as my own thoughts, in this post while not including major spoilers when possible. I'll also include Thrawn in this post since I think there is a certain, yet lesser, degree of confusion in regards to him as well. Darth Vader does not really have an exact position within the Imperial ranks. In most cases, he is regarded as the Emperor's main enforcer and thus he has default authority over practically all Imperials. Tarkin is sort of an exception, depending on the scenario. Whenever the two are assigned together, Palpatine explicitly states to the two who will be the mission leader. But regardless of the case, Tarkin is off-limits to Vader when it comes to execution, as Palpatine explicitly stated not too long after the start of the Empire. This is because like Anakin, Tarkin was specifically chosen by Palpatine years before the Clone Wars and conditioned to become the ruthless tyrant he is destined to be. So Tarkin is considered "precious cargo" to Palpatine. And yes, Vader was tempted to strangle Tarkin in at least one instance. The two's relationship did not start off well but it improved over time. In one instance, Vader asked Palpatine how he views Tarkin in comparison to him. Palpatine responded that the Sith is more important to him than the Empire, and Vader being his Sith apprentice is of the most importance to him. Thus, in the grand picture, Palpatine values Vader more than Tarkin, but he does not outwardly show it as no other Imperial is informed that the Emperor is a Sith Lord. (source: the Tarkin novel) Bringing Thrawn into the picture. Thrawn is a most unusual case in a lot of aspects (his non-human species and his origins being examples). In fact, Palpatine did not know of Thrawn's existence until well after the Empire was formed and thus did not start off being Palpatine's brain child as Vader and Tarkin did. But it was actually because Thrawn was an alien from the Unknown Regions that he struck Palpatine's favor. Thrawn offered a service to the Emperor that no other follower, including Tarkin and Vader, could offer: knowledge of the Unknown Regions that would allow for the Empire to expand even further. Palpatine saw Thrawn's value, and so accepted him into the Imperial Navy (after a couple months of academy training). Thrawn rose through the ranks at an unprecedented speed, both due to his special circumstances and his military genius. After a mere couple years, Thrawn was already Grand Admiral. Thrawn and Tarkin have a good relationship. In terms of who outranks who, there is no definitive answer as the two belong to different branches. Ultimately, I would personally say Tarkin has more authority than Thrawn but there are arguments for both sides. In Rebels, Tarkin set up a meeting between Thrawn and Palpatine to discuss funding issues. While this indicates that Tarkin has certain privileges with the Emperor that Thrawn does not, looking at the Thrawn novels this may not necessarily be the case. Palpatine and Thrawn do indeed meet frequently to discuss expansion into the Unknown Regions as well as addressing Thrawn's personal questions (the Death Star being one of them). So Tarkin setting up the meeting may have been more along the lines of unburdening a friend's list of tasks by doing one of them for him rather than granting a request that the person could not achieve otherwise (as in the case of what Krennic asked of Vader in Rogue One). Why I ultimately chose Tarkin (and again, I may be wrong) is because Palpatine knew Tarkin a lot longer. He also seems to trust Tarkin a bit more than he trusts Thrawn, as seen by how Thrawn remained unaware of the Death Star's existence until he was already a Grand Admiral (but that is not to say that Thrawn isn't entrusted with information kept from Tarkin, especially when it comes to the Unknown Region expansion). If there was a hypothetical scenario where one would want to punish the other, I would think that neither would have the authority unless granted by the Emperor himself since both serve critically important roles. And yes, officials normally have the authority to execute/punish someone of lower stature, even if it's someone in a different branch (military vs. politics and governorship in this case), as seen by Tarkin's execution of Aresko and Grint and Thrawn's implied planned execution of Pryce. Thrawn and Vader do not have as good a relationship. The two were assigned together in Thrawn: Alliances, which took place shortly after Season 3 of Rebels. Like Tarkin, Thrawn was off-limits to Vader and in this case, Vader was seriously tempted to disobey his master as Thrawn pushed Vader's patience to a great degree several times. The main reasons for the tension between the two are the following. First, Vader is aware of Thrawn's recent failure with the rebels and he is not happy about it. Second, Thrawn possesses information that he is not obliged to tell Vader (as the Emperor himself stated), and Vader cannot stand that another individual aside from his master does not follow his orders. This also made Vader suspicious of Thrawn's true loyalties (the Empire or the Chiss), as these situations were usually revolved around knowledge of the Unknown Regions. Palpatine was initially skeptical of Thrawn's true loyalties in this regard as well but has since been cleared up. Third, Thrawn actually worked with Anakin back in the Clone Wars. Other than that experience serving yet another reminder to Vader of his past life, Thrawn figures out that Vader and Anakin are the same and implies it at one point (and of course, Vader rebukes it, and Thrawn is like okay, whatever you say). So with Thrawn in the picture, I would say that the three form an unofficial triumvirate as the Emperor's top enforcers, where each individual holds official authority over the others only when the Emperor says (this kind of triumvirate is similar to the one of Kylo Ren, General Hux and Captain Phasma). P.S. While on this subject, I'll briefly elaborate on Grand General Tagge, an event from the 2015 Vader series. After the first Death Star blew up, Vader was disgraced for his failure. To punish him, Palpatine promoted Tagge (one of the Death Star officers who was not present when it was destroyed) to not only Grand General, but as someone who Vader answers to. This was done merely to humiliate Vader, and does not speak for either Tagge's competence or his actual value to the Empire (as was in the cases of Tarkin and Thrawn). By the end of this spiel, Vader redeemed himself, and wasted no time in killing Tagge after the off-limits ban was lifted.
He was still around to sign the Empire's surrender after the Battle of Jakku, at least (book 3 in the Aftermath trilogy). After the surrender, basically he ruled Coruscant for a while, but under the supervision of New Republic officials.
I assume that Vader is pretty much the "voice of the Emperor" - he's outside of the hierarchy proper, he has no "official" rank politically but can step in and slap down pretty much anyone who ISN'T The Emperor regardless - he was SPECIFICALLY told he now did answer to Tagge, implying he did not before (and only temporarily in any case). It just so happens that people like Tarkin he has a good relationship with and shares political views so they mostly happen to align anyway. To use an example - Tarkin had no power/authority to MAKE Vader stop force-choking Motti - but he decided to only after he'd made his point, a few seconds AFTER Tarkin says enough.
