This is a fun question! 361 was of course quite an important year, since at the end of the year Julian the Apostate became the emperor, the last non-Christian to be at the helm of the Roman Empire. Rome however was no longer the lone heart of the Roman world, as Julian continued his march on Constantinople and was acclaimed there instead. Rome still had value, so Julian did take care to address the Senate respectively and duly appointed the necessary magistrates, but he did not visit the Eternal City. Luckily, the main historian of this period, Ammianus Marcellinus, also wrote about another imperial visit to Rome - Constantius II's adventus (his ceremonial entrance into the city) in April 357. Ammianus was writing around 391, so it is possible that the Rome he described also partially reflected the Rome he lived in, but his words are still nonetheless impressive:

So then he [Constantius] entered Rome, the home of empire and of every virtue, and when he had come to the Rostra, the most renowned forum of ancient dominion, he stood amazed; and on every side on which his eyes rested he was dazzled by the array of marvellous sights. He addressed the nobles in the senate-house and the populace from the tribunal, and being welcomed to the place with manifold attentions, he enjoyed a longed-for pleasure; and on several occasions, when holding equestrian games, he took delight in the sallies of the commons.

This was of course all very traditional: the Senate was still to be respected, the people received an address from their rightful emperor, the races were still held.

He also surveyed the great public buildings of Rome:

He thought that whatever first met his gaze towered above all the rest: the sanctuaries of Tarpeian Jove so far surpassing as things divine excel those of earth; the baths built up to the measure of provinces; the huge bulk of the amphitheatre, strengthened by its framework of Tiburtine stone, to whose top human eyesight barely ascends; the Pantheon like a rounded city-district, vaulted over in lofty beauty; and the exalted heights which rise with platforms to which one may mount, and bear the likenesses of former emperors; the Temple of the City, the Forum of Peace, the Theatre of Pompey, the Odeum, the Stadium, and amongst these the other adornments of the Eternal City. But when he came to the Forum of Trajan, a construction unique under the heavens, as we believe, and admirable even in the unanimous opinion of the gods, he stood fast in amazement, turning his attention to the gigantic complex about him, beggaring description and never again to be imitated by mortal men.

These buildings were of course all very famous, though even then there would have been plenty of churches, most of which were built further away from the traditional centre of the city. There were still many pagans in Rome and their hold on the urban space had yet to be contested (this would have to wait until the time of Pope Damasus, who embarked on a mass church-building programme). Constantius at this point also removed the Altar of Victory from the Senate House, though the statue of Victory remained. The altar was possibly later restored by Julian, so if you were a visitor to the city in 361 you may well encounter something that had not been seen for years. Furthermore, to celebrate his own victory over an usurper five years previously and the twentieth year of his reign, Constantius also erected an obelisk in the Circus Maximus. The obelisk was brought all the way from Egypt in emulation of earlier efforts by Augustus and Constantine I and it was still standing by the time of Ammianus. Similar obelisks soon sprung up in Constantinople and elsewhere in Rome - copying the emperor was after all very fashionable.

All in all, Rome in 361 was still an impressive city, particularly if we also draw upon Ammianus' descriptions of other events. Ammianus detested both the lazy lower classes and the degenerate rich, but his descriptions of the decadent nobles' superficial lives and the urban poor's willingness to riot over wine are enough to paint a picture by themselves. It was still a bustling city and (generally) well-supplied with grain and wine, a city with a population of around 800,000, a far-cry from the austere philosophical utopia preferred by Ammianus, but an impressive sight nonetheless. Indeed, a Persian prince who accompanied Constantius on his adventus had this reaction to the city:

When Ormisda was asked directly what he thought of Rome, he said that he took comfort in this fact alone, that he had learned that even there men were mortal.

Rome in 661 on the other hand was very different. It had experienced two 'barbarian' sacks of the city in the fifth century, three sieges in the Gothic War of the sixth century, and was regularly affected by plague, famine and more sieges by the Lombards in the decades leading up to 661. It was therefore much smaller; Chris Wickham suggested that it had 1/20 of the population of Rome in the fourth century, so c.40,000. Rather than a city, it could perhaps be seen as "a set of what could be called urban villages, maybe as many as a dozen, held together by a common politics and, probably, a continuing ritual of processions across the old classical centre." There were also innumerable churches and our best source for the city in this period, The Book of Pontiffs, a collection of papal biographies, repeatedly recorded endless church-building projects by various popes. Famous buildings like the Pantheon had also been converted to churches, though in this case quite late, only in 608. There were still new additions to the Forum however, such as the Column of Phocas dedicated to the emperor Phocas (602-610), but large-scale secular building projects had stopped.

My interest though is in church history, and in that sense the city was as exciting as ever. Due to the last war with Persia between 603-28 and the Arab conquests from the 630s onwards, eastern refugees flooded west. Rome quickly became a hub for these travellers, with distinctive Armenian, Palestinian and North African monasteries all attested to. People from even further afield can also be found there, my favourite being the cult of Anastasius the Persian, a Persian martyr who was executed in 628. There were also many pilgrims, even from as far away as Anglo-Saxon England, with quite a few kings deciding to end their lives in Rome after setting aside their crowns. Infrastructure was also in place to support these visitors. Though much diminished, Rome still had a special place in the hearts of many Christians.

661 also proved to be a decisive year in the long-running war between the Roman Empire and the Arab Caliphate. In that year the Umayyad caliph, Mu'awiya, emerged triumphant from the first Arab civil war and soon an offensive on all fronts was launched against the empire. Resources were swiftly mobilised by the Romans to counter this, with one event being particularly strange to modern eyes - the emperor Constans II began to prepare to sail to Italy instead of marching east. At the time though this was a very strategic move, as Italy, especially Sicily, was still relatively prosperous and provided much-needed funds for the emperor. It was also an excellent place to check Arab advances into North Africa and a place to rebuild the shattered Roman navy, an arm of the military key to any efforts to defeat the Arabs in this new 'Mediterranean world war'. Moreover, Rome was of dubious loyalty to the empire, as during a previous crisis, the war of 649-54 against the Arabs, the papacy in fact rallied Italy against the emperor, partly due to the emperor's doctrinal position. Constans managed to reassert his authority then, but the papacy was still not particularly trustworthy. Thus in 663, having arrived in Italy the previous year, Constans entered Rome, the first emperor to do so for nearly two centuries:

The Apostolicus went to meet him [Constans] with his clergy at the 6th mile from Rome and welcomed him. On the same day the emperor travelled to St Peter’s for prayer and there he presented a gift; <another day to St Paul’s and there he presented a gift;> on Saturday to St Mary’s and again he presented a gift. On Sunday he proceeded to St Peter’s with his army, all with wax tapers, and on its altar he presented a gold-wrought pallium; and mass was celebrated. Again on Saturday the emperor came to the Lateran, and bathed and dined there in the basilica of Vigilius. Again on Sunday there was a statio at St Peter’s, and after mass was celebrated the emperor and pontiff bade farewell to each other. He stayed in Rome twelve days; he dismantled all the city’s bronze decorations; he removed the bronze tiles from the roof of the church of St Mary ad martyres, and sent them to the imperial city with various other things he had dismantled.

As you can see from the itinerary, the emperor spent his time visiting churches rather than the secular buildings Constantius had visited. The city was still the source of much wealth, but at this point it was in the hands of the Church rather than the senatorial aristocracy. Constans desperately needed cash to fund his war and as this also conveniently weakened the papacy, this was in fact a very astute move. The papacy however had been growing in influence for decades and this was only a temporary setback. It was an institution deeply enmeshed in the government of the city, in welfare and of course was the only thing available to provide spiritual sustenance for the inhabitants. Rome was no longer a city dominated by the secular state, as it was in 361, but now by the Church. The city may have been patrolled by imperial soldiers in 661, watchful of any seditious papal behaviour even as the crisis of empire deepened, but the loyalty of the city was now leaning towards the pope in Rome, rather than the emperor in distant Constantinople.

To add to this: How did fashion change in this time for the average Joe Titus and Arria Albius in those 300 years?

For one, Rome was still a very much in use city in 361. Declining since the move of the capital some decades earlier but still lively. In the mid-late 5th century is when the Vandals sacked the city. By this point, due to several several factors (inflation, immigration, lack of administration etc) a lot of the population had left the city. Rome was largely symbolic. When the Vandals sacked the city (and it wasnt just an invasion, this included people who were local inhabitants), they took everything. They stripped the copper from the rooves. They raided homes and businesses. Rome lost everything. This was after multiple raids that had taken place in the last two years. The population was under 35,000 by this point, down from the nearly million it had been approximately a century before.

In the 6th century, Belisarius mounted a campaign to remove Ostrogoths from Rome, and successfully defended it for a couple years. According to the accounts written of it, the city was broken down and mostly uninhabited. They used cracks in the aqueducts to sneak into the city. Unfortunately, the emperor was paranoid that Belisarius would be too powerful and arrogant and try to seize the throne so he left him undermanned, eventually allowing Ravenna and Rome to be retaken.

For centuries it passed between multiple peoples until the Lombard invasion. It was Byzantine in name only, but they had no administrative control over it. So it would be a very very marked difference.

Source: Lost to the West by Lars Brownworth

Edit: hit reply too early