This questions asks for your intuition and insight as I'm surprised by how little is known about the difference between nonnegative and positive curvature. I don't want to be completely vague, so I could ask: What are the difficulties and currently blocked paths to solving the Hopf Conjecture? (Does $S^2\times S^2$ support a metric of positive curvature?). But in general, I would like to know what others might know on why it's difficult to determine if a given closed simply-connected space of nonnegative curvature can also admit positive curvature. As far as I know, there are no obstructions, how come? The amount of examples of nonnegative curvature compared to that of examples of positive curvature seem to suggest there should be something distinguishing the two.