"A large amount of those 1.3 billion people are still working in the countryside on farms. Whereas out of the total population of the U.S., only 2% work on farms. China was not aided by the U.S. in industrialization as Japan was, plus they started a great deal later. It is unfair to compare a 3rd-world country's economy to a very well developed one."
It isn't unfair because a third world country is a third world country on principle; most African nations for example are in such horrid conditions not because of 'fate' or some other mystical reason, but because few (if any?) countries in Africa have ever had protected individual rights, specifically property rights.
China has failed to develop faster because their political system has kept them stagnant economically for a hundred years (particularly the period where Mao reigned). Most of China (the farm lands that you reference for instance) is ruled in a feudal agrarian manner, by lords that preside over a territory by consent of the communist leadership in Beijing.
When America went through industrialization, it didn't have the benefit of existent industrial technology and knowledge that could help speed up the process dramatically; China has that benefit, but their political system has held them back a great deal (although they've learned to be far more lenient than the Soviets were, or else their economy would get choked off to an extreme point (as it did with some of Mao's adventures in communism)).
China is being aided by the U.S. in its goal of further industrialization, and it is being aided by Japan to a great extent (that is, to the extent that China will allow either to invest in the country to help speed industrialization). China was not aided much previously because they wouldn't allow investment, or they wouldn't guarantee not to nationalize the investments (ownership stakes, new companies, etc.), or they would place so many restrictions on investment that it wasn't worthwhile to invest (such restrictions remain, but to a lesser extent).