Magnetic bearings eliminate all mechanical parts of conventional bearings. They operate contact-free and need no lubrication since the rotor is kept floating by a magnetic field.
Contrary to the conventional ball/roller bearings there is no more mechanical friction and no wear and tear. In addition, by using electromagnets, an active magnetic bearing can actively counteract vibration of the rotor, thereby decreasing noise and increasing the lifespan of the remaining engine components. Since it´s almost friction-free, this technology would possibly help reducing the fuel burn even further (subtract from that the energy required to operate the bearings).
Without mechanical wear, magnetic bearings could also have much longer lifespans and longer maintenance intervals.
So in principle, the advantages are pretty impressive. If they can actually make it work, reliably...
Problems could be in added complexity and possibly weight (although ball bearings aren´t exactly lightweight, either).