The reason was simple: Strategy, and the needs of government.
Rome was, simply put, very far from the frontiers of the empire. It was not possible to coordinate military campaigns on the Rhine or the Danube, let alone in Syria or Anatolia, from Rome itself.
Rome had and continued to enjoy tremendous symbolic and ideological importance in the later empire, but it was of little strategic import. The true wealth of the empire came from the provinces, particularly rich ones like Egypt and Africa, and as a city Rome likely took in far more money than it produced for the state, as it literally ate up the surplus of the empire in the form of the giant grain fleets.
Rome was also hard to defend. It was poorly positioned, it did not have a naturally dominant and easily defensible position like Constantinople did.
The later capitals of the western Empire were Milan, which was much further to the north and therefore much better situated to rule from, and Ravenna, which was not only further north but also located in a supremely defensible position in the middle of a series of swamps.
Now, as to your second question: I have no idea whether any emperor felt it was dishonourable to move the capital, since only Marcus Aurelius was kind enough to leave us a proper diary and his rule pre-dates the change, but I very much doubt it.
You see, the nature of the Roman empire had changed during the crisis of the third century, and its century of gravity had shifted long before any official attempts were made to move the capital. For fifty years, civil war wrecked the empire and invasions and raids disrupted life along all the borders while emperor after emperor was murdered or killed in battle. The situation was finally salvaged by a generation of military officers from the Danube frontier, skilled military men, often from relatively humble backgrounds, quite unlike the Italian and provincial aristocrats who had ruled the empire in its heyday.
Many of these men hardly set foot in Rome during their reign. Famously, the great Diocletian, who completed the period of recovery and in many ways set the empire on the path it would stay on for centuries, only visited Rome once, (probably) and did not like it. Meanwhile, cities on the frontier such as Trier had gained great regional importance and prosperity, simply because they were near the imperial court much more of the time. Milan gained prominence in this fashion as early as the reign of Gallienus in the 260s, as the base for his field army.
Rome's size continued to be unrivalled by all but a handful of places in the empire, but its political importance was a thing of the past. By the time the last generations government of the Western Roman Empire found it necessary to move their seats of government to Milan or Ravenna, it was not a shocking event. When Alaric sacked Rome in 410 A.D. it was a tremendous shock to the Roman world, but it was a symbolic rather than a strategic one, a symptom and not a cause of the changes that had come over the Empire since the days of Caesar and Augustus.
As for what impact this had on Rome itself: a few years ago, u/Shlin28 wrote one of my favourite posts on the sub on the topic of Rome: Suppose you visited Rome in AD361 and then AD661, what differences would you see? It should be enlightening and entertaining.
Sources:
Goldsworthy, The Fall of the West
Heather, The Fall of the Roman Empire