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.
Yau asked in 1982 if there is any compact simply connected manifold with nonnegative curvature for which one can prove that it does not admit a metric of positive curvature. This question opens his list of unsolved problems in geometry (see "Seminar on Differential Geometry", p. 670.)
Let me quote from "A Panoramic View of Riemannian Geometry" by Berger (Springer 2003, p. 579):
A major difficulty is that it is not clear how to find the critical set of the sectional curvature (as a function on the set of tangent planes).
The earlier short survey by Bourguignon contains a discussion of the reasons why some seemingly natural approaches fail.