Julian the Apostate campaign in Persia is a complete success with all of his goals being met. But he dies suddenly after his campaign. What results is that Christianity is restored as the state church by his successors but they don't have to give up some holdings on the eastern boarder like in otl. Does anything in Roman history changes after this?
This topic his funny is funny because my first Real TL was about a, relatively, successful Persian Campaign.
The Romans don't really have to be successful against the Persians, all they need to do is to go back to Roman territory with their armies intact and that would change much.
But let's go to your scenario.
First the problem would be where Julian dies. If he dies while with the Army, they are bound to chose one of their own leaders as Emperor, but if he dies after returning to Antioch, shit is bound to happen.
In the first scenario you could see a more experienced officer than Jovian to be chosen as Emperor. Another point is that OTL the office was offered to the Perfect of the Orient Salutius, who refused it on the grounds of being old and having bad health. Another point is that if we consider that Ammianus Marcellinus is correct and Jovian was elected incorrectly then the office might go to the chief notary, Jovianus. These are the choices if the army as all the saying.
But in Antioch you have too many possibilities with every men with enough money trying to bribe it's way to the purple. This could easily cause a civil war, that would be more destructive than OTL failed Persian Campaign.
But let's imagine that everything goes well to the Romans and that a experienced men is chosen to the Purple.
Their first act is to choose a successor, and given that we neither know of any sons from either Salutius or Jovianus, then that men must be a outsider that they trust and he must have connections to the army. A possibility could be Theodosius the Elder (The father of Theodosius the Idiot, the destroyer of the West, the son of a ***** that should had been drowned at birth... you get my drift) that was by all accounts a loyal and brilliant general. Another possibility could be one of the sons of Gratian the Elder, either Valentinian or Valens, he was a very popular men among the Army and having him as an ally could reinforce the claim of the new Emperor, given that he would have all reasons to support the new Emperor if his son was bound to gain the throne.
This is just a small part of the politics your scenario could create.
On the Religion part it depends what kind of Christian sect the new Emperor follows, is he a Arian or not? That could have many effects on the way Christianity evolves.
On the rest everything would change little.
Rome was still overstretched with too many enemies on too many borders, a successful Persian campaign puts the Sassanids out for a while but they are bound to seek revenge and on the west the Franks, Allemani, Quadii, etc... are all of them looking for a way to gain more lands and riches at the cost of the Empire, you would need, at least, some three good Emperors, both reigning for long times, to change the fate of the Empire, and the intact army would be a bonus for those Emperors, so lets avoid Hadrianopolis and Theodosius the Not so Great and everything could go fine (as long as no idiot reaches the purple, but that is asking too much).