Did the West ("Rich Countries") win from the Rest of the World? Depending on the period you talk about it's true. With hindsight of historical data of Gross Domestic Product (in Dutch 'Bruto Binnenlands Product') it's only true between 1850-2010. Before China and India were bigger. And if the predictions are valid .... they will be in the future too. GDP data between year 0 and 2050 for nine superpowers:
Source: NRC Handelsblad. April 4th, 2015
For the biggest part the historical data above, of the long term economic growth, is collected by Angus Maddison (1926-2010).
Some Americans talk about the "Old World" when they refer to Western Europe. I guess it's better to point at China and India when we talk about the Old World. Western Europe and it's offspring (read: USA) are the newbies and in betweenbies.