This is a 10,000 foot answer to a broad question, but it often comes down to whether the colony served as a producer for the metropole or served as a consumer for the metropole. I want to emphasize, no colony fits perfectly in any one category, and there are major arguments that can be made on a more local level. Broadly speaking, though, territories that attracted permanent settlement of families and developed organic political structures fared better than those that were formed in the interests of resource extraction and attracted lower numbers of largely elite men to manage disenfranchised native populations.

The former type of colonies were less valuable to Britain as economic producers, but extremely valuable as markets for British goods. As a result, these peripheral colonies were largely left to manage themselves and developed organic political and social leaders. With these colonies, particularly the United States, Canada, Australia, and New Zealand, the large numbers of British colonists meant that they recreated British institutions of local governance and legal structures.

The latter colonies primarily existed to benefit the British economy (self-righteous paeans to a "civilizing mission" not withstanding) and to enhance Britain on the global stage. These colonies were drawn with resource extraction first in mind. Thus, after the dissolution of the Ottoman Empire, you wind up with nations like Iraq, where the borders make no organic sense to the native populations but do make sense to BP. This is the same in Africa, where the colonial / national borders have much more to do with the claims made by European colonizers than the ethnic realities on the ground. Beyond problems with borders, these extraction colonies had deliberately limited introduction to British legal traditions and liberal education. The role of the natives in these colonies was to assist with extraction, not to learn how to govern themselves. Infrastructure also served imperial ends. Rail lines, roads, bridges, etc, were built to conduct resources out of the region, not to improve the larger society. So, when the British finally left, they left an infrastructure that ensured oil and mining companies could continue to profit, but people in the cities and villages would be left without basic social needs. Leadership fell to the few that were well educated and/or wealthy. Without an organic political system that represented everyone in the new nation to adopt, power concentrated in a few hands or fractured along traditional ethnic / religious lines.

Colonies like the American South and South Africa represent a blurring of those two categories. They were more extractive than consumption-oriented, but had large enough Anglo-European populations to drive investment in local structures and politics. Thus, while they eventually became liberal democracies (sort of), it took generations fighting the exploitive nature of their societies.

I'd love to have a historian of colonial India chime in, because I feel it's something of a unique case. It had a strong imperial tradition before the British arrived, with well-defined institutions that survived colonial pressures (though were changed by them). At the same time, the partition was awful, and the resulting antipathy between Muslim and Hindu Indians is a worrisome legacy.