Just going to throw this out there, it seems to me that the Pope is primarily the successor of St Peter, which the Bishop of Rome is by definition, but how exactly are they one and the same thing? Stay with me here.
When Our Lord gave Peter the keys, and said "upon this rock", Peter was not yet in Rome. Neither was he in Rome at Pentecost. Rome is where he ended up, and where tradition holds that he died. Peter has to have a successor, so Linus becomes that successor, ergo Bishop of Rome, ergo Pope. What other way would there be to choose a successor to St Peter, than to have him be Bishop of Rome?
I suppose the question is, whenever a Pope is chosen, is he first and foremost chosen to be successor of St Peter, and being so chosen, becomes Bishop of Rome and thus Pope, kind of ex officio, if you will? Or is he elected Bishop of Rome by proxies for the Roman clergy, viz. the College of Cardinals, and having been thus elected, is Pope ex officio, and being Bishop of Rome/Pope, is regarded as the successor of St Peter?
Or are all three things "one big ball of wax" happening simultaneously?