Question

how did the foreign policy doctrines of the two bush presidents differ

Answer 1

The Bush Doctrine grew from neoconservative dissatisfaction with President Bill Clinton's handling of the Iraqi regime of Saddam Hussein in the 1990s. The U.S. had beaten Iraq in the 1991 Gulf War. That war's goals, however, were limited to forcing Iraq to abandon its occupation of Kuwait and did not include toppling Hussein.

Many neocnservatives, and indeed many Americans in general, were unhappy that the U.S. did not depose Hussein. Post-war peace terms also dictated that Hussein allow United Nations inspectors to periodically search Iraq for evidence of programs to build weapons of mass destruction (WMD), which could include chemical or nuclear weapons. Hussein repeatedly angered neo-cons as he stalled or prohibited U.N. inspections.
Neoconservative Letter to Clinton

In January 1998, a group of neoconservative "hawks" (who advocated warfare, if necessary, to achieve their goals) sent a letter to President Clinton calling for the removal of Hussein. They explained that Hussein's interference with U.N. weapons inspectors made it impossible to gain any concrete intelligence about Iraqi weapons. For the neo-cons, Hussein's firing of SCUD missiles at Israel during the Gulf War and his use of chemical weapons against Iran in the 1980s erased any doubt that he would use any WMD.