Simply put, Barack Obama is against the use of torture and has specifically banned it in his administration, while Romney is for the use of "enhanced interrogation" techniques, or what most of us see as torture (waterboarding is a good example).
Barack Obama Quotes on the Use of Torture
In 2007, then Sen. Barack Obama (after his run for the presidency had been announced) issued a press release titled ”Torture and Secrecy Betray Core American Values”:
“The secret authorization of brutal interrogations is an outrageous betrayal of our core values, and a grave danger to our security. We must do whatever it takes to track down and capture or kill terrorists, but torture is not a part of the answer - it is a fundamental part of the problem with this administration's [George W. Bush] approach. Torture is how you create enemies, not how you defeat them. Torture is how you get bad information, not good intelligence. Torture is how you set back America's standing in the world, not how you strengthen it. It's time to tell the world that America rejects torture without exception or equivocation. It's time to stop telling the American people one thing in public while doing something else in the shadows. No more secret authorization of methods like simulated drowning. When I am president America will once again be the country that stands up to these deplorable tactics. When I am president we won't work in secret to avoid honoring our laws and Constitution, we will be straight with the American people and true to our values.”
On January 22, 2009, Barack Obama issued Executive Order 13491 which banned the use of torture and ensured lawful interrogations:
(a) Common Article 3 Standards as a Minimum Baseline. Consistent with the requirements of the Federal torture statute, 18 U.S.C. 2340 2340A, section 1003 of the Detainee Treatment Act of 2005, 42 U.S.C. 2000dd, the Convention Against Torture, Common Article 3, and other laws regulating the treatment and interrogation of individuals detained in any armed conflict, such persons shall in all circumstances be treated humanely and shall not be subjected to violence to life and person (including murder of all kinds, mutilation, cruel treatment, and torture), nor to outrages upon personal dignity (including humiliating and degrading treatment), whenever such individuals are in the custody or under the effective control of an officer, employee, or other agent of the United States Government or detained within a facility owned, operated, or controlled by a department or agency of the United States. […..]
Mitt Romney Quotes on the Use of Torture
During a GOP primary debate on May 15, 2007, Romney gave lots of love to Guantanamo Bay, and said that “enhanced interrogation techniques have to be used”, although he claims that “torture” and “enhanced interrogation” are two different things:
“Now we're going to -- you said the person's going to be in Guantanamo. I'm glad they're at Guantanamo. I don't want them on our soil. I want them on Guantanamo, where they don't get the access to lawyers they get when they're on our soil. I don't want them in our prisons. I want them there.
Some people have said, we ought to close Guantanamo. My view is, we ought to double Guantanamo. We ought to make sure that the terrorists -- and there's no question but that in a setting like that where you have a ticking bomb that the president of the United States -- not the CIA interrogator, the president of the United States -- has to make the call. And enhanced interrogation techniques have to be used -- not torture but enhanced interrogation techniques, yes.”
In an Associated Press article dated July 20, 2007, Romney repeated the refrain that he supports “enhanced interrogation” but not “torture” (Romney aides have said that Romney does not believe that waterboarding is torture.)
“I support tough interrogation techniques, enhanced interrogation techniques, in circumstances where there is a ticking time bomb, a ticking bomb... I do not support torture, but I do support enhanced interrogation techniques to learn from terrorists what we need to learn to keep the bombs from going off.”
After a GOP primary debate on Nov. 11, 2011, Romney aides talked about Romney’s positions on torture and killing/targeting American citizens working with the enemy:
Romney aides said after the forum that he [Romney] does not believe waterboarding is torture. His adviser Eric Fehrnstrom would not say whether the practice would be used in a Romney administration, saying decisions about "enhanced interrogation" will be up to the president."He is not going to spell out what he would employ", Fehrnstrom told CNN. Romney embraced several hawkish positions and said military forces should be able to target and kill American citizens fighting alongside enemies overseas, even without a trial.