It is unclear what is meant by "philosophy of Islam", since
Islamic philosophy did not really engage Shiite Islam; it was
primarily done by Sunni Muslims.
If the question is asking about religious differences, the key
really lies in successorship to the Prophet Mohammed, both in his
earthly domains (i.e. the ruler of the Islamic State) and his
spiritual qualities (i.e. his perfection as a person and instrument
of God). The Sunnis believe that revelation ended with Mohammed and
that the Islamic States should be ruled by qualified individuals
chosen in any viable mechanism. The Shiites believe that revelation
continued after Mohammed through the lips of the Infallible Imams
(of which there were 5, 7, 11, or 12, depending on which form of
Shiite Islam a person practices) and that only those appointed by
God from Mohammed's family have the right to rule Islamic
States.