The contest involves "soft power" as China seeks raw materials to fuel economic expansion, growing influence on world currency markets, and more dangerous brinksmanship, evident in cyber attacks on the West and in sabre-rattling against anxious neighbours, including Japan and the Philippines, allied to the US.
"There is a high level of distrust existing, and even growing, between the two countries," says Richard Solomon, a senior fellow at the Rand Corporation.
The cyber attacks and brinksmanship in the China Sea raise concerns, he says, "about a drift towards confrontation".
On June 7 and 8, the US and China will try to defuse tensions - or as the White House puts it, "discuss ways to enhance co-operation, while constructively managing our differences" - when President Barack Obama and China's new leader, Xi Jinping, meet in Rancho Mirage, California. The over-arching issue is how the US and China can accommodate each other in a bi-polar, or multi-polar world that challenges US hegemony.
Washington sees a more aggressive China, investing in greater maritime capability, missile technology and cyber warfare.
Beijing views the US as a power in decline. America is still the world's biggest economy, but China is catching up and, according to the OECD, will be top dog by 2016.
The first Obama Administration shifted military assets to the Asia-Pacific region as part of a "pivot," an effort interpreted by Beijing as a brazen form of containment.
It also sees the Trans-Pacific Partnership Agreement talks - which include New Zealand - as another bid to exert US influence in the region.
Can next week's summit reduce tensions?
"We have a new Secretary of State. China has a new leader," says Elizabeth Economy, a senior fellow at the Council on Foreign Relations. "Is this an opportunity to improve the relationship? There's cautious optimism from many in the US. I am not terribly optimistic."
China is famously opaque. An AidData analysis, conducted by the Centre for Global Development and the College of William and Mary in Virginia, found China spent about US$70 billion ($87 billion) on "soft power" projects, such as sports stadiums and scholarships, in Africa between 2000 and 2011 - compared to US$90 billion from the US - but Economy says in By All Means Necessary: How China's Resource Quest is Changing the World (co-authored with Michael Levi) that overseas investment can be less than it seems.
While China is the world's largest exporter and the largest lender to developing nations, she say the gap between rhetoric - what is promised - and reality is sometimes wide. While Beijing has created dozens of export zones often "there's virtually nothing there," except a Chinese construction company and investor. "You don't have any investors, Chinese or otherwise, interested in setting up in these zones."
Moreover, cosying up to regimes such as Burma or Zimbabwe could "earn the enmity of the broader population". China is also more wary of high-risk investments. Who will provide security once Nato quits Afghanistan?
Great Game comparisons stand less scrutiny when Beijing's shortfalls are taken into account. Historically, China has been an inward looking nation. This is a handicap in the globalisation era.
The US is seasoned at diplomacy and global deal-making; China is a relative novice. Its grand offer to help broker peace between Israelis and Palestinians last April - prompting fears China would fill a vacuum if the US pivot reduced engagement in the Middle East - lacked specifics on how it would do so.
For while China is an economic titan, the power this bestows is sometimes shallow. China lacks diplomatic experience, crucial for working with the international community. It can wield a bigger stick, pace its growing blue water navy and cyber skills, but it has yet to learn to speak softly. China "still lacks the clout to bang heads together" in the Middle East said the Economist, suggesting Beijing would remain on the diplomatic sidelines.
Abiding by former leader Deng Xiaoping's "hide and bide" dictum - "adopt a low profile and never take the lead" - no longer works for China, as the US and others ask it to step up and help shoulder global governance.
"The Chinese now seem to be feeling their way towards doing that," says Economy, but they are hampered by a long tradition of disengagement with the wider world and by a paucity of skilled diplomats. "So you find the same people running climate change negotiations, that are involved with the G20, and who are trying to deal with the South China Sea."
China's growing power is more broad than deep, a handicap that invites missteps - such as the row with Japan over the Senkaku islands. While this appeases Chinese nationalistic sentiment, it also heightens regional instability, crucial for the economic growth that helps Beijing keep the lid on civil unrest.
Fathoming China's intentions and aims can be challenging. Solomon suggests one problem may be "a disparity between what the civilian leadership and the military is doing".
Is the tail - the Peoples Liberation Army - wagging the dog? He cites the Bo Xilai scandal, and Beijing's alarm that the disgraced politician had close ties to the PLA and, according to the Communist Party, was courting military support.
More than 50 years ago, with Britain on the wane as a global power and the US ascendant, Harold Macmillan wondered if the UK might mentor the US, "Athens to America's Rome". It was an arrogant miscalculation.
But as China takes to the world stage, engagement with the US could help Beijing work out how to be a global player. A code of conduct, creating mechanisms to defuse tensions so both powers talk to each other, not past one another, on vital issues would be a start.
When Secretary of State John Kerry visited Beijing in March, Xi Jinping's public welcome said Sino-US relations were "at a new historical stage", floating the idea of a new cooperative relationship with Washington. Kerry said he wanted to "understand the roadmap ahead".
It remains to be seen if both powers will share the same road, rather than face off in a competitive Great Game that may ultimately benefit neither.
Two nations
China
• Area: 9,596,961sq km; world's fourth largest country
• Population: 1,349,585,838, median age 36.3 years
• Capital: Beijing, 12.214 million
• GDP: US$12.38 trillion
• Labour force: 795.4 million
• Military spending: 2.6 per cent of GDP
United States
• Area: 9,826,675 sq km; world's third largest country
• Population: 316,668,567, median age 37.2 years
• Capital: Washington, DC, 4.421 million
• GDP: US$15.66 trillion
• Labour force: 154.9 million
• Military spending: 4.6 per cent of GDP
Source: CIA Factbook
