Obama and others in the White House who support Summers were deeply impressed by his ability to navigate the financial crisis. Administration officials who endured the tumultuous crisis period -- and many people who didn't -- assume the next crisis is a matter of when, not if. Obama has confidence that Summers would be effective in handling such a crisis, while he barely knows Yellen.
And if the financial players know you believe this, and if they know "we" have all learned the supposed lesson that you can't let any too big too fail institution fail, then they know that they can steal all the money and wait for the free money water cannon to be pointed at them.