There is a lot of discussion about how the "Obama Doctrine" differs from the "Bush Doctrine." While I'm not sure that it yet qualifies as a “doctrine,” and seems like more of a case-by-case approach, the Obama administration's foreign policy in many respects is certainly quite different than the Bush approach.
The “Bush Doctrine” sought expansive ends, such as the democratic transformation of an entire region and the defeat of global terrorism; and it authorized preemptive war as means to these ends. The Obama administration has yet to articulate such a grand strategy.
Further, Obama's foreign policy decisions are being made in a very different international context. The US position in the world is quite different than when Bush launched wars in Afghanistan and Iraq.
To begin with, the US is mired in an economic slump, and weary of war. This constrains foreign policy decision making. The American public is highly skeptical of spending money on military intervention abroad, when there are pressing financial concerns at home. Furthermore, the wars in Afghanistan and Iraq have made Americans far more skeptical of military intervention (public support for the mission in Afghanistan is only around 40%). Thus, broad ideological aims, such as "democratization," are far less likely to be seen as in the "national interest."
Perhaps more importantly, however, because of the wars in Afghanistan and Iraq, the US is already stretched militarily. Thus, preemptive action is less of an option than when Bush articulated his "doctrine," in 2002. Although US military capabilities are unmatched, they are not infinite.
Further, France and Britain dragged the US into a shooting war instead of the reverse. This is also partially due to the fact that the US is already heavily committed militarily. However, by seeking to enhance the legitimacy of the UNSC, rather than contesting its legitimacy, Obama seems far more committed to multilateralism than Bush.
More importantly, the US is effectively reacting to events in the Middle East, rather than catalyzing events there. The Obama administration is making decisions based upon actions by the people of in Egypt, Libya, Syria, Yemen, etc.; rather than forcing their governments to react to US actions.
For these reasons, and others, the Obama foreign policy approach is inherently different than the Bush approach. It is more multilateral, and far more reactive rather than proactive. The grand-strategy of the Bush administration was transforming the world in US interests; however, Obama's seems to be to preserve US interests in a changing world.