This has to do with a concept called "colonies of settlement vs. colonies of exploitation." In the United States and many other former British colonies, the primary goal was settlement. In other words, the native populations were displaced and wiped out, only to be replaced with settlers from the mother country. These settlers not only shared the same language, values, religions, and other institutions as the mother country, but they were broadly accepted as more-or-less cultural equals back home. They were also allowed to build infrastructure and lasting societies. When the time for independence came, these countries were more readily accepted as full members of the international community.

Contrast that with the pattern of European colonization in Africa (and largely in British India and much of Southeast Asia), in which small European elites presided over a large population of enslaved and/or disenfranchised indigenous peoples, largely for purposes of resource extraction. Rubber in the Congo, slaves on the Atlantic coast, and various precious gems and metals in various places were taken out of their home countries, usually by native labor held by coercive force. The elites did not consider themselves inhabitants of a new country, but rather temporary residents who hoped to escape back home once they had enriched themselves. Once these nations finally gained independence they had not been permitted to build infrastructure and social institutions, and they were not readily welcomed into the international community of nations.

Sources:

Wallerstein, World Systems Theory

Patrick Wolfe, “Settler Colonialism and the Elimination of the Native,” Journal of Genocide Research 8, no. 4 [December 2006], 388

Veracini, Settler Colonialism

[This Guardian Article] (http://www.theguardian.com/politics/2005/aug/20/past.hearafrica05) also has some good stuff in it.