Before revision: The major distinction among the three is the ratio of Josephson energy vs (Coulomb) charging energy: For phase qubit, this ratio is on the order of 106 , whereas for flux qubit it's on the order of 10, and for charge qubit less than 1.
After revision: The major distinction among the three is the ratio of Josephson energy vs (Coulomb) charging energy: For phase qubit, this ratio is on the order of 106 , whereas for flux qubit it's on the order of 10, and for charge qubit less than (which allows for macroscopic bias current through the junction)