That raises an obvious question: Why does British Prime Minister David Cameron have to ask legislators for permission to attack Syria while Obama doesn't?
The short answer is that Cameron technically didn't have to get Parliament's consent, but he asked anyway. Obama, by contrast, is arguably required to seek approval from Congress — but might not. (Update: Obama announced Saturday that he'll ask Congress to vote on a Syria strike after all — though he insisted he didn't need permission.)
Why Cameron asked permission: In Britain, the prime minister can technically declare war on his own under the royal prerogative — that is, under powers delegated from the monarch to the prime minister. But ever since 2003, when Tony Blair decided to consult with lawmakers before sending troops into Iraq, prime ministers have generally sought Parliament's approval.
That custom prevailed in the Syria debate — even though this was the first time a prime minister failed to get permission for military action. After the vote, Labor leader Ed Miliband asked Cameron to promise not to use the royal prerogative to strike Syria anyway. Cameron agreed: "I can give that assurance," he said. "I strongly believe in the need for a tough response to the use of chemical weapons, but I also believe in respecting the will of this House of Commons."
As it happens, British politicians have occasionally debated a resolution that would formally hand war powers over to Parliament. Gordon Brown proposed the idea (pdf) in 2007, and Cameron himself was a big supporter of the idea. But so far, there's been no rush to pass the bill, in part because prime ministers have made a habit of asking lawmakers for their consent on war matters.
Why Obama might not ask permission: Obama is technically supposed to get authorization from Congress to go to war. More precisely: Under the War Powers Resolution of 1973, the president is required to notify Congress within 48 hours of committing armed forces to military action abroad. He then has 60 days to get approval from Congress; otherwise, he has to cease operations within 30 days.
But that didn't happen when Obama sent warplanes to Libya in 2011. The administration argued that the War Powers Resolution didn't apply so long as the United States wasn't involved in "sustained fighting or active exchanges of fire with hostile forces" or sending "U.S. ground troops." (Instead, the U.S. was dropping bombs from planes and firing missiles.) This argument provoked plenty of internal disagreement: Obama reportedly rejected the Office of Legal Counsel's view that he needed authorization from Congress.
The Libya decision was controversial at the time. Yale legal scholar Bruce Ackerman warned that "Mr. Obama’s decision to disregard [the Office of Legal Counsel's] opinion and embrace the White House counsel’s view is undermining a key legal check on arbitrary presidential power."
Now the same argument is popping back up with Syria: The Obama administration says it has the authority to strike on its own, and Congress is divided. A few members seem willing to grant some leeway here. Sen. Robert Menendez (D-N.J.) said the president has "certain powers even under the War Powers Act that he can use [in] the national interest of security, and he can act.” (Menendez did, however, ask Obama to explain his rationale to lawmakers.)
Others are standing up more forcefully for congressional authority. At least 98 Republicans and 18 Democrats in the House have signed a letter calling on the president to seek formal approval for military action. "Engaging our military in Syria," it says, "when no direct threat to the United States exists and without prior congressional authorization would violate the separation of powers that is clearly delineated in the Constitution."
It's also worth noting that back in 2007, then-candidate Barack Obama seemed to agree with the ask-Congress-for-permission camp, telling the Boston Globe: "The president does not have power under the Constitution to unilaterally authorize a military attack in a situation that does not involve stopping an actual or imminent threat to the nation."
So how does that fit with President Obama's current stance? The White House is now arguing that the alleged use of chemical weapons in Syria — in violation of international norms — does in fact constitute an "imminent threat." Here was White House spokesman Jay Carney earlier this week: “Absolutely, allowing the use of chemical weapons on a significant scale to take place without a response would present a significant challenge to or threat to the United States' national security interests."
Related: There's much, much more historical background on the use of military force and the role of Congress in this report (pdf) from the Congressional Research Service.
Update: On Saturday, Obama announced that he'll ask Congress to vote on a Syria strike, after all — though he insisted that he didn't need congressional authorization