People have been trying to define China's role on the global stage for a lot of years. Having read a big chunk on the subject, this recent riff from the Washington Post's Ezra Klein got my attention:
"Americans don't necessarily like funding the military and aid expenditures that go along with being the world's single superpower, but they're pretty into the idea of being that superpower. That's why the conflict with China that's implied in American political rhetoric is that they're going to 'catch up' to America, or they'll have the fastest train and we won't. They have more people, faster growth, a bigger class of freshly-minted engineers, and so on. To the American mind, it stands to reason that they'd want to make a play for the crown.
And maybe they will, eventually. But the more common policy conflict between America and China right now is that we want them to step up and do more on the international stage and they want to be left alone to do less so they can focus on domestic development. Afghanistan, global stimulus, global warming, Iran, and North Korea all follow this pattern. The Chinese phrase for their approach is 'Keeping a Low Profile and Taking a Proactive Role When Feasible,' and they're putting a fair amount of effort into arguing for it."
While this might oversimplify a story of mismatched interests, values, and capabilities, I think as far as 20,000-foot view summaries go, it hits the nail on the head.
If you want to really get into it, read what Stephen Goldsmith and Daniel Wagner have to say. I don't agree with the conclusion, but it's a good issue-packed diatribe nonetheless.