One advantage will be finally, that Galileo will use a constellation, which is allowing better coverage of Earth, but requires more satellites.
Also, it will start with Hydrogen-Maser-clocks, while GPS is slowly phasing this technology into the system when satellites get replaced.
Next, Galileo will provide a better coverage of the polar regions, because all satellites operate at a higher inclination as GPS.
Disadvantages are simpler: It is not 100% compatible to GPS and it will take a while until it is ready. It will also cost money for more than the basic services, but these high quality service are harder to get for GPS.
And another disadvantage: It is not yet decided who will really operate Galileo. In the worst case, it is a European consortium of these companies who are already responsible for delaying the program.