Actually the city of Rome never really ceased to be the symbolical centre of the empire. That explains for example the shock that was felt by many throughout the empire when Rome was captured by Alarich in 410.

But there definitely was a decline in the importance of the city as an actual seat of political power that began already before the time of Diocletian. In the early 3rd century the Roman historian Herodian (1,6,3-5) wrote:

Rome is where the emperor is

which shows two things:

• With the establishment of the monarchy under Augustus not the city of Rome and its institutions but the emperors were thought of as the sole focus of political authority.

• the word "Rome" could be used as a metaphor for the seat of power separately from the actual city.

In the times of the soldier emperors of the 3rd century the monarchs were most of the time away from Rome because they were fighting foreign invaders or usurpers in the provinces or were murdered before they could reach the city. There are already examples for joined rulership of more than one emperor from this time like with Valerian and his son Gallienus. So one could say the empire was already beeing divided into several halves. But those didn't have their own capital cities.

This kind of changed with Diocletian and the establishment of the tetrachy. The emperors now chose cities in the provinces like Trier or Thessaloniki as their residences and heavily invested in them by building large palaces, victory monuments, baths etc. Those cities were even compared to Rome. But they weren't real capitals. They were still part of the provincial system unlike Rome with its own praefectus urbi. And while the emperors visited them more often than Rome they were still away for most of the time. Also one emperor often had more than one important residence like Constantius Chlorus with Trier and York. For the whole of the 4th century the emperors traveled from city to city and frontier to frontier. The situation could be compared to that in the Holy Roman Empire of the Middle Ages with its traveling monarchs and lack of anything like a real capital.

It gets a little more complicated when one comes to the foundation of Constantinople by Constantine the Great. Often one can read that Constantine shifted the capital of the empire from Rome to his new foundation. That wasn't really the case, because as already stated the Roman Empire didn't really have a capital at the time. Constantinople was more akin to the other imperial residences of the time although it quickly overshadowed them in terms of importance and imperial building projects.^1

With the beginning of the 5th century the emperors begann to "settle down again". Constantinople became the permanent seat of the eastern emperor making it the de facto capital of the east. Western emperors now often resided in Ravenna which is why that city is often called the last capital of the Western Roman Empire. That is problematic because in the 5th century Rome was for the first time since the 3rd century used as an imperial residence as well. So one could make a case for Rome and Ravenna beeing joint capitals in the 5th century, at least of the soon to be collapsing Western Empire.^2

So how did all of this effect the city of Rome? As already mentioned it kept much of its symbolical importance. The emperors of the tetrarchy still came to Rome to celebrate the anniversaries of their accessions. So did Constantine (at least twice). It was always a large event for an emperor to visit Rome like Ammianus Marcellinus describes for Constantius II. and Theodosius I. Rome was still the seat of the senat. Although it wasn't required any more for senators to actually live in Rome since the days of Constantine, many still did. And those that did were more often than not the ones with the most illustrious line of ancestors and the biggest estates in the provinces. Public offices from the days of the republic like the consulate were still appointed and their holders donated lavish spectacles for the city at the days of their accession.

Rome was also still the largest city in the Roman Empire and arguably the world. Its massive population was still supported by the annona, the continued shipping of large quantities of foodstuffs to feed the citizens of Rome. That only began to change in the 5th century, when the city was plundered at least three times and the province of Africa, where most of the corn that was supposed for Rome originated, was lost to the Vandals.

I don't know of any evidence that this development did damage Roman prestige or pride in any way. For most of its citizens the empire still remained the res publica with the city of Rome as its center. The establishment of the monarchy at the end of the 1st century B.C. hadn't changed that and the fact that those monarchs now resided somewhere else didn't either.

1. G. Dagron: Naissance d'une capitale: Constantinople et ses institutions de 330 à 451 (1974)

2. A. Gillett: Rome, Ravenna and the Last Western Emperors, in: Papers of the British School at Rome 69, 2001, 131-167