Yes they were the western Roman Empire, however since Contantine had moved the capital to Constantinople the concept of the empire being tied to the city declined. Being "Roman" wasn't about having the actual city of Rome anymore. That said, it still held a great deal of prestige (the senate continued to meet there until at least the 600's AD). In some ways, you could almost think of it like the Hague and Amsterdam. The capital was moved for several reasons. Mediolanum was further north, enabling the western emperor to reach the German frontier more quickly when necessary. It was a major hub for the roads going north and south. Mediolanum was chosen mainly because it was far enough north to still have access to the frontier while still being in Italy. The Emperor Honorius moved the capital to Ravenna because it was a very well-defended city and the Roman frontier had effectively collapsed. For an emperor with not much hope, this city was the perfect choice. It had been a Navy base in times past and still had good access to the sea, as well as rough terrain and a strong wall surrounding it. It should be noted, however, that the decision to move the capital to Ravenna in many ways contributed to the eventual sack of Rome. Hope this helps!

It would probably be helpful to contextualise when and why Milan (Mediolanum) became the 'capital' of the WRE.

The middle of the 3rd century is typically called the 3rd century crisis because it was a time of massive disruption in Roman history, with significant economic collapse, pressures on the borders, and a string of emperors who could barely hold power for more than a year. During this period the city of Rome didn't far so well itself, and began to decline in importance.

In 284 Diocletian came to power, and the great legacy of his rule was to essentially bring stability and order back to the empire. Diocletian to a large extent sidelined the importance of Rome and the Senate as no longer that important to Imperial rule. He also instituted the Tetrarchy system, and around 285 appointed Maximian to co-emperor in Milan. This became the strategic and leading city in the WRE from this time onwards (though not always); in the East, Diocletian tended to base himself in Nicomedia.

We tend to think of Rome as the most important place in the Roman empire, and there's a certain truth to that, in the Classical and Hellenistic periods. But Rome turned from a city, to a city-based empire, to a regional empire ruled by an/several 'emperors', in which period the importance of Rome as a city declined, and so did its population. Late Antique emperors never based themselves in Rome, and although their visits to the city were symbolically important, they made little of the city itself and did not invest their time and their power base there.