Short answer: Obama rebounded strongly in this debate. Romney was prepared and did as well as he did in the first one, but Obama's performance was so much better than last time, he gets the win.
Mitt Romney got the first question (from a college student asking about the availability of a job after he graduates). He walked over to "Jeremy" and spoke directly to him. Barack Obama also walked to the student and used the answer to attack Romney on wanting to "bankrupt Detroit."
Candy Crowley busted the rules right off the bat by asking a follow-up question - what about short-term jobs - which both men used as an excuse to get on each other.
Twice in the first two questions Obama said "Governor Romney's plan …" and then described it. Romney missed the opportunity, at least in the early going, to say "I'll talk about my plan, why don't you tell us what you plan to do?"
Obama speaking directly to Romney - "you stood in front of a coal plant and said 'this plant kills' now you're a big coal guy." That will make Obama's supporters very happy.
Twenty minutes in, both are doing well, both are making their points, both are performing up to their capabilities.
At 23 minutes in, Romney backed Candy Crowley down and answered a charge by Obama in spite of her trying to effectively say, if you want to use up your two minutes on the next question that's up to you. He didn't so he kept going.
Romney's answer on the question of taxes on middle class families (whom Romney referred to as "middle income" families) was just about perfect. Obama's answer was good, too (98 percent will see no tax increase, but the GOP is holding them hostage to protect the tax rates of the top two percent.
The Obama team has obviously given him one fact with which to attack Romney on each of the major issues - jobs, coal, and taxes so far. Good technique because he doesn't look like "Belly Laugh Biden," but he is making his points. As an example, Obama's attack on Romney during the follow-up on taxes was very effective.
Question from young woman about pay disparity went to Obama first and he talked about his mom - very good approach, but he ran out of stuff before he ran out of time so he pivoted to Pell Grants ("to make education easier for women") which seemed to be off-point.
You don't HAVE to use up all your time on every question.
Romney's answer was surprisingly facile - including an anecdote about reaching out to women's groups to staff his cabinet and senior staff as Governor. He pivoted to unemployment and how it affects women.