It is actually false that the Shrub used September 11th
directly as an excuse for the invasion of Iraq. Cheney was cleverly used to make the implication, while leaving the Shrub the "plausible deniability." The Shrub did not ever actually say that Iraq was involved in the September 11th attack--but capitalized on the vague belief common among all too many ignorant Americans.
Even before Hussein entered politics in Iraq, Iraq was a secular state, dominated by Sunni Muslims who were repressing the Shi'ite majority, and making casual war on the Kurds with the connivance of the Turks, who really, really hate the Kurds. The Ba'ath Arab Socialist Party was secular, but primarily based among the Sunnis of Iraq (and Syria, the other state in which they became powerful). The leadership of al Qaeda are Wahhabis, who are fundamentalist Sunni extremists, and they have cordially despised Iraq and the Ba'athists. Before the Gulf War, when al Qaeda were still our buddies, before American troops in Saudi Arabia gave bin Laden his excuse for war on Americans, bin Laden even offered to settle Hussein's hash for us.
But Americans are ignorant, don't know a Sunni from a Shi'ite, and don't know anything about the Wahhabis. It is a simple matter for the conservatives to whine about "islamo-fascists," and reduce a complex matter to simplistic terms, which allow them to hate all Muslims without giving any serious thought to the matter.
At Wikipedia:
The Wahhabis
The Sunnis
The Shi'ites