Both of them - "il principe" and "'o principe" translate to the prince in English. What is the difference between the prefixes here?
Italy has a peculiar history with respect to its official language. In the 14th century, the great writers Dante, Petrarca and Boccaccio established a literary form of the Tuscan vulgar language which was adopted as a written language by the learned people.
There were several different languages spoken by people in the different states Italy was divided into, some of them with a literature. The language spoken at Naples spread over most of Southern Italy, which was a single state under Naples rule (however, Sicily and the southern part of Calabria never adopted Neapolitan).
It was not until 1860 that, due to unification of Italy, Italian became the official language all over the new state. What's Italian? It's Tuscan, mostly, but the different languages influenced each other during the long cohabitation.
In standard Italian, the article for principe (prince) is il. But in other Italian languages it can be different. In Neapolitan, the article would be ’o (which is an apocope of lo); in Venetian it would be el; in Sicilian it would be lu or ’u; in Sardinian, su.
Your ‘o principe is not standard Italian, but Neapolitan. Two different languages, which share the word for prince (remember about the mutual influences of the Italian languages). You wouldn't ask about le vase (French) and the vase (English), which both translate into il vaso in Italian, would you? 🙂
A better example might be der Bach (standard German) and de Bach (Swiss German): two different, but cognate, languages sharing a word with the same spelling.