"I know a guy" might get you a job or back stage passes for a concert in the US.
"I know a guy" can get you off of vehicular homicide in China.
Granted, the US still has an occasional taste of the second, but it's far less common.
http://www.businessinsider.com/wal-mart-billionaire-alice-wa...
Being good friends with the right folks at city hall will make sure your company wins the contract to do the landscaping at the city park.
Knowing the right people will generally get any permit you need expedited or at the very least it won't be delayed.
If special treatment is corruption then corruption is very much alive and well in the US. The difference is that the image of corruption is corruption is much less tolerable. If the details of the story of you running over someone makes the news or someone points out the conflict of interest of the person awarding the contract the corruption won't happen in that case. "I'm sorry I can't do that" is a much less common phrase than "I'll put in a good word for you".
Corruption in US (federal, state and local) is a lot like the restaurant that only cleans up its act for the health inspector. Government is the restaurant and publicity is the inspector. When there's a spotlight on the problem everyone plays by the rules. When nobody's looking it's business as usual.
Acts of obvious corruption (trading favors, bribes, etc) is rarer in the US than some other countries but leveraging relationships for special treatment is just as common as anywhere else, we just keep it under the table.
This isn't an attack on government but government is certainly not the consistent and impartial organization that some people seem to think it is (though I think that's a good goal to have).
Edit: I find it odd/interesting that whenever I say something that isn't positive about government it gets up-voted a couple times then down-voted a couple times. I really should write a script to track it because I have a hunch that the timing follows a particular pattern.
In the US, you have an army of groups who will actively/passively work against a corrupt person: free media, lawyers, civilians with strong rights, judges, police, opposition politicians, civil liberties groups, government oversight committees, industry regulators, consumer groups, and generally speaking, a law-abiding and non-corrupt population.
In China, you have none of those.
However bad you think corruption in the US is (and I believe you when you say it's bad), it's degrees worse in China.
So it encompasses business networking, social favors, family favors, but also nepotism, political corruption, perjury, bribery etc.
So if the guy at the print shop gave me a good price after my firm helped him out the week earlier with his broken air conditioning, then that's guanxi. If I managed to illegally obtain environmental certificate for my horrendously polluting plastics factory because my brother works at the environment department, that's also guanxi.
And even when it's to get hired, the "guy" had better be able to do the work...