Do Italians say "la cattedrale di Milano" or "il duomo di Milano?" Or both? How are these two different?
A cathedral or cattedrale in Italian, is just the Church were a bishop has his cathedra (essentially, a fancy chair -- think throne, but for bishops). Any church can be a cathedral, if the bishop so chooses.
Duomo is a customary name for the main Church of the city. It is often also the cathedral, but not necessarily so: for example in Venice before the Napoleonic conquest the cathedral was the church of San Pietro in Castello, which was not the main church of the city (that would have been St Mark's basilica, which is the current cathedral). For another example, the main church of Rome is St Peter's basilica, but despite what one would think, it is not the city's cathedral, that's the church of San Giovanni in Laterano.
The normal name for the Church of Santa Maria Nascente in Milan is Duomo di Milano. It is also the cathedral of the city, and so referring to it as the Cattedrale di Milano would not be incorrect, but unusual. It seems it is used more often on the website of the Dioceses of Milan (as one would maybe expect, since what's relevant there is the role played by the Duomo).
So, to sum it up:
- Cattedrale: the bishop's seat.
- Duomo: an informal designation for an important church.
- Basilica: a church which has been given particular cerimonial rights by the pope.
Exactly by which name any particular church is designed varies widely.