Great question, though I would like to point out that we can't really describe Christians as Roman Catholics in the fourth century, since all Christians would call themselves 'catholic' (and orthodox for that matter). It is an inappropriate term to use simply because there wasn't an institutional divide between the east and the west by that stage, even though there were myriad religious disputes. Instead, it is better to divide Christians into different theological/ideological camps, like Arian (a broad term, but it will do generally) and Nicene, rather than to see them as separate Churches.

With the conversion of Constantine in the early fourth century, the Pope certainly became more influential, but he did not exercise any political powers - that would come in the sixth century when Italy was reconquered by the Eastern Roman Empire and the Papacy was delegated several important duties, particularly during the reign of Gregory the Great. Within the Western Roman Empire, I would suggest that the Pope's role was heavily dependent on the Pope's own ability and his personal relations with stronger individuals like the emperor, rather than by virtue of his office.

For example, it is evident from fourth century sources that the pope wasn't actually that independent. From Ammianus Marcellinus, our strongest contemporary source, the Papacy was only a position to be fought over by power-hungry men, rather than a position of renown or anything particularly holy. Two candidates for the Papacy, Damasus and Ursinus, urged their followers to fight against each other after a contested election, leading to hundreds of deaths in just one basilica and the urban prefect/governor of the city fleeing to the suburbs because even he couldn't control the rioting. These papal candidate sound more like the corrupt nobles Ammianus described rather than particularly spiritual men.

Then again, Ammianus was a pagan, so he perhaps would describe Christians this way. Some historians however would say that Ammianus was being quite accurate and objective about this (in the same chapter he praised virtuous provincial bishops for example). The disputed election was after all a continuation of the struggle between the popes Liberius and Felix under Constantius II, who supported one then the other in order to maintain stability - Constantius II had favoured the Homoian brand of Arianism, which for many Nicene Christians was a bad thing, hence the imposition of his favoured candidate in order to quell dissent. By the time of Damasus and Ursinus, religious disputes were not the priority any more, as Valentinian I wanted stability rather than strife (as compared to Constantius, who regularly deposed bishops and tried to impose his will on the Church), hence Valentinian's toleration for chaos in Rome during his reign and the lack of punishment for the people involved when stability returned. For Valentinian, as long as the Church chose someone, he didn't care that much. In both cases, the Papacy was not a strong institution, as it was either a tool to enforce the emperor's will or a way to bring about internal peace. In broad terms, other bishops closer to the imperial court, such as Ambrose of Milan, were more important in actually affecting high politics (at least, that's what our sources, Ambrose's letters, would have us believe...).

Popes were still influential figures of course, as when Damasus finally became the pope, he did try to make major changes. He sponsored Jerome to create a new translation of the Bible and also expanded the network of churches in Rome, reshaping the contested urban landscape (Rome still possessed a vigorous pagan aristocracy at the time, so this was a big deal!). Later, popes would play a major role in condemning heresy (against Pelagianism for instance) and settling religious disputes elsewhere (at the Council of Chalcedon in 451, the judgement by Pope Leo the Great was key to its final decision). Leo was a particularly interesting figure, as he also negotiated for the retreat of Attila the Hun during his invasion of Italy, so on the surface, the Papacy was seemingly more assertive.

However, looked at more closely, the Papacy was not in such a strong position. Jerome's papal-backed Bible translation wasn't popular with North African congregations (they preferred the traditional translation they were used to), whilst Pope Zosimus' condemnation of Pelagianism only came about due to imperial pressure from Emperor Honorius (the Papacy was ambivalent to Pelagianism beforehand). Leo's intervention at Chalcedon was an exception, possibly only because he was a particularly energetic pope. Other prominent western churchmen, such as St Augustine, were in fact snubbed by eastern bishops, as they were seen as uncultured - they didn't even know Greek! As for his dealings with Attila, it is debateable to what extent Attila was going to retreat anyway for logistical issues and how strong a negotiation position Leo really had.

tl;dr: the pope was an important figure when he had the opportunity, but the Roman Empire was a large and dangerous place filled with powerful men and although the Church was increasingly influential, he was still the underdog when he interacted with the secular elites of the empire.

Sources:

• J. R. Curran, Pagan City and Christian Capital: Rome in the Fourth Century on Damasus' reshaping of Rome

• Any articles/books by Peter Brown or Robert Markus would be a good if you are interested in the broader Christian culture of Late Antiquity, The Rise of Western Christendom by the former and The End of Ancient Christianity by the latter are particularly excellent

• Ammianus' account of the disputed papal election can be found here