What's the difference between a cathedral and a basilica?

The references I've seen so far have alternated primacy between one and the other.

Solution 1:

A basilica was originally, in Hellenistic Greece, a tribunal administering justice on behalf of the king (βασιλέως - basileus). The word and the thing were adopted by the Romans after annexing Macedonia in the 2^nd Century BC. The archeological site of Pompeii, for instance, has the remains of a large basilica which served as a tribunal, and a few hundred meters away in the modern city of Pompeii a catholic basilica.

When Constantine officially legalised the Christian faith (Edict of Milan) as one of the Empire’s state religions (4^th century A.D.)¹, he erected a number of large churches which were then named basilicas. The current Basilica of St Peter’s in Rome replaces a previous building built precisely by Constantine. As a general rule basilicas are the highest ranking churches and are declared so by the Pope. Some high profile basilicas are built on top of the burials of notable saints (St Peter's, St Francesco of Assisi, St Vasily in Moscow, St Paul Outside the Walls...)

Cathedrals, on the other hand, were typically built during the Middle Ages. They are headed by a bishop. However, if this bishop is head of his diocese, then his cathedral outranks any basilica in his diocese.

Neither cathedrals nor basilicas are restricted to the Catholic faith: see St. Basil's in Moscow, of the Russian Orthodox Church (remarkably sometimes translated as basilica; sometimes to cathedral).

¹ He also decided we would rest on Sundays ;-)

Solution 2:

A cathedral is the proper term a church that is home to a bishop. A basilica may refer to anything from a church's architecture to its importance to the pope, depending on its type. The Holy Roman Catholic Church categorizes basilica according to their function: palace, a papal seat of authority, etc. Some cathedrals are also called basilica, but it's generally the case that a basilica outranks a cathedral in matters of church authority.

Solution 3:

A cathedral is the principal church of a diocese, while basilica is the name given to certain churches granted special privileges by the pope.

Solution 4:

A cathedra (Latin, "chair", from Greek, kathedra, "seat") or bishop's throne is the chair or throne of a bishop.