How Is the Bush Doctrine different from the Obama Doctrine?

The phrase Bush Doctrine was first used by political commentator Charles Krauthammer in June 2001 to describe the Bush Administration's unilateral withdrawals from the ABM treaty and the Kyoto Protocol. The phrase initially described the policy that the United States had the right to secure itself against countries that harbor or give aid to terrorist groups, which was used to justify the 2001 invasion of Afghanistan.

Unlike the Monroe Doctrine, the Obama Doctrine is not a specific foreign policy introduced by the executive, but rather a phrase used to describe Obama's general style of foreign policy. This has left journalists and political commentators to speculate on what the exact tenets of an Obama Doctrine would be

The terms "Bush Doctrine" and "Obama Doctrine" as they are commonly discussed are probably misnamed. The term "Bush Doctrine" was invented by Bush opponents during the Obama/McCain campaign and those that invented it apparently meant it to mean Bush's War in Iraq, his economic policies, his social ideas on issues like stem cell research, abortion, etc. Similarly, the Obama Doctrine is more directed at Obama's general philosophy and policies.
This can be better understood by comparing them to the Monroe Doctrine or the Truman Doctrine. These are doctrines that define how the United States aligns itself with the rest of the World.
It might be better to talk of Bush's policies versus the Obama policies. That makes it much easier to answer the above question.
Bush's policies favor the free market (capitalism) while Obama's policies favor Government control of the market (socialism) and redistribution of wealth (communism). If you don't like the terms socialism and communism being used when describing Obama, maybe some clarification is in order.
A common definition of socialism is when the Government runs industry.
And communism is best described by Karl Marx in his communist manifesto as:
".....from those according to their abilities .... to those according to their needs.... "
or in the words of Obama, "Spread the wealth."