John is right to note that Obama gets more slack than Bush on some of his counterterrorism policies – namely, most if not all of the elements of the aggressive Bush policies that he continued and (in some respects, like drones) ramped up. This is in part explained (as Peter Baker, quoting Jennifer Granholm, suggests) by NGOs and Democrats (and perhaps others) trusting President Obama more than President Bush when he acts aggressively (and by their related desire not to want to criticize President Obama). I also think that it is explained in part by Obama’s quite different rhetoric, as compared to the first-term Bush administration. Whatever the underlying reality, we have heard no interviews or speeches by senior Obama officials proclaiming an administration priority to expand presidential power, and we have heard more self-constraint and rule-of-law rhetoric. (To be clear, John never engaged in such self-defeating rhetoric while in office and in my experience did everything he could to fight it.) The rhetorical differences do not make matter a lot, especially over time, but they matter a bit, I think.
For the sake of completeness, we should also note that on some policies, President Obama gets much less deference and less room for maneuver than President Bush. I am thinking mainly of criminal trials for terrorists in the United States, and discretion to release GTMO detainees. Few criticized Bush as being “weak on national security” when he did these things. And yet Obama has been criticized practically from the beginning on these issues, and has found himself severely constrained by Congress. Many things explain this asymmetry, including political opportunism (which also applies to the asymmetry noted in the last paragraph as well). But an important explanation is that just as Obama the “civil libertarian” is trusted more than Bush on aggressive counterterrorism measures, Bush the “cowboy” is trusted more than Obama on “soft” counterterrorism policies.
I discuss these asymmetries in Chapter 2 of Power of Constraint. They are among the complex implications of the “Nixon Going to China” phenomenon. A classic academic treatment of the subject is Robert Goodin, Voting through the Looking Glass, 77 Am. Pol. Sci. Rev. 420 (1983) (behind JSTOR paywall).