As I understand it, there were two reasons for leaving Saddam in power after 1991.

The first was that the UN mandate was strictly about expelling Iraqi forces from Kuwait.

The second, as I understand it:

1. The Saudi government and population are Sunni muslims.
2. The Iraqi government is Sunni/Secular.
3. The majority of the Iraqi populace is Shiite.
4. Iran is mostly Shiite.
5. There is a history of animosity between the Sunni muslims and the Shiite muslims.

Deposing the Iraqi government would have left Saudi Arabia with a majority Shiite country right on their northern border. This would have made Saudi uncomfortable. Leaving the sunni/secular government in charge in Iraq would have been a better option from the Saudi point-of-view.

Moreover, Iraq served as a useful buffer between the fundamentalists (again Shiite) in Iran and Saudi.

I guess that the US administration figures that:
a) Iran is more moderate these days.
b) They're next, anyway. cf Axis of Evil.
c) They'll be replacing one (tyranical) secular goverment in Iraq with another (non-tyrannical) secular government.