Drew has been one of those QB's who has had to learn how to win along the way. In New England he passed aggressively but they didn't always have the defense and running game (beyond those precious years with Curtis Martin) to support him. Thus, Bledsoe ended up trying to do too much.
He took it as his responsibility, and so we are used to watching him try to force things in the passing game to win. But last year he was able to forgoe the pass-first approach and hand the ball off to McGahee. Most people forgot how well he did that in NE with Martin, too. And now he's still willing to do what the coaches ask of him.
Some QB's, like Dan Marino and for most of his career John Elway, either never got the opportunity to play with a balanced team or truly believed they could do it all with their arm. Bledsoe has confidence, but he has learned that he can't win games by himself.