Article I, Section 9, clause 2 of the Constitution: “The Privilege of the Writ of Habeas Corpus shall not be suspended, unless when in Cases of Rebellion or Invasion the public Safety may require it.” For the first objection, we were facing neither rebellion nor invasion when Bush suspended this right, but Lincoln clearly had rebellion on his hands.
Lincoln made a public declaration that he was suspending Habeas Corpus. You can read it here.
When Bush suspended it, he made no such declaration. He just did it all sneaky-like, and even that was only after he was caught. In the Supreme Court case Boumediene v. Bush, Bush’s suspension was found unconstitutional. You can read Justice Kennedy’s majority opinion, Souter’s concurrence, and Scalia’s dissent in the Wiki article.
What it all hinges on is the Constitutionality of the Military Commissions act of 2006. This is the legislation that Bush rammed through Congress that allowed him to do pretty much whatever he wanted. In this case, one of the things he did was suspend Habeas Corpus for people already detained, which is an ex post facto law (a law that criminalizes something retroactively). It also defined “enemy combatant” far too broadly, to the point where (as one example states), if you had ever donated money to an orphanage in Afghanistan, the President could have hauled you in as an enemy combatant.
You can read more about the differences in Lincoln’s and Bush’s treatment of Habeas Corpus here.
In short, Lincoln had a clear reason and performed transparently and according to the Constitution. Bush did not meet the Constitutional criteria of “rebellion or invasion” (many of the Gitmo detainees were not even arrested on U.S. soil) and did so pretty obscurely and far too broadly.