I was explaining to someone my upcoming research project into the sources of the earliest collections of papal letters recently, and this question came up. It’s actually a very good question, because it helps clarify what the person with whom you are speaking actually means by the terms. There is a certain kind of Roman Catholic, for example, who would say that not only are ‘Bishop of Rome’ and ‘Pope’ synonymous, the office of the Bishop of Rome has pretty much always been invested with the same authority and whatnot.
My answer was that, for my research, I use the terms interchangeably. However, it is more that I mean ‘Roman Bishop/Bishop of Rome’ when I say ‘Pope’ than that I mean ‘Pope’ when I say ‘Roman Bishop/Bishop of Rome’. That is, I am conscious of a development in the office of the Roman Bishop and his role in ecclesiastical polity that means that ‘Pope’ Siricius (d. 399) and ‘Pope’ Innocent III (d. 1216) and ‘Pope’ Francis do not all have exactly the same job or role in the wider church.
John Moorhead’s 2015 book, The Popes and the Church of Rome in Late Antiquity takes the same tack, although Moorhead eschews the adjective ‘papal’ and noun ‘papacy’, with good reason. I choose not to because they are short words and everyone knows what they mean; instead, I frame my use of pope-related words at the beginning of my work so people know what is going on. Calling Leo ‘Pope’ is perfectly legitimate; therefore, talking of his papacy makes a lot of sense to me — although I can also see Moorhead’s perspective, trying to avoid clouding the issue of how the Roman Bishop’s role developed.
What is a ‘pope’? A ‘pope’ is a papa in Latin — a father. The term is used in the fifth century of bishops beyond the Bishop of Rome, although eventually it becomes restricted to said bishop in its usage. I am fairly certain no one ever legislated the term ‘pope’. It is also used in Eastern churches; hence the current Pope Tawadros II of the Coptic Orthodox Church. At St Sozomen’s Church, Galata, Cyprus, it is painted in a fresco on the exterior of the building; the fresco is of the Council of Nicaea, and the poor fifteenth- (sixteenth-?) century artist knew neither Pope Sylvester’s name nor the fact that he wasn’t even there, so simply painted ‘Papa Romis’ over his head.
It is a title of honour, originally used to esteem the person and activity of spiritual direction of the bishop. Therefore, even though Bishops of Rome in the late 300s and 400s were not the same sort of Pope as Innocent III, they are still Popes — and they still claim a primacy of honour. And Pope Leo the Great, in fact, even claims that all clerical ministry descends from Peter, and therefore Rome.
How the pope, in his role of Bishop of Rome, Metropolitan of Suburbicarian Italy, and holder of a primacy of honour, Patriarch of the western church, comes to be invested with universal jurisdiction and appoints all bishops is a different story. But to call someone ‘pope’ need not imply said jurisdiction or vision of the papal role.