This is a very broad question, and the answer depends upon the definition of several terms.
First, "British" refers specifically to Britain, i.e. the United Kingdom, which was formed in 1707 following the merger of the kingdoms of Scotland and England. Prior to this, the North American colonies were specifically English. Thus if we're talking about "British colonization" we're limiting ourselves to the post-1707 time period. However, I'm going to assume we're talking about both English and British colonization.
Asking which is the most influential factor in colonization is like saying "which is more important, eating or drinking"? Some might personally favor one or the other, and one might be of great importance in certain circumstances, but in the end they are inextricably linked, and ascribing superior importance to one is impossible because we can't quantify the effects of either. Basically, if you removed either religion or economics, you wouldn't have ended up with the same colonies, but we don't know what kind of colonies you would have gotten either.
I think the answer depends on the time period you're considering, and how closely you look. In very broad terms, religion was more influential, whereas economics were the more practical and everyday motivators.
English colonization arose from a combination of factors, most of them directed against Spain. First, England and several other countries protested against the idea of Spain claiming sole dominion over the New World, and especially against the Pope literally awarding the continent to Spain exclusively. This was one of many complaints against the Catholic church, which grew into the Protestant Reformation. This made England, a Protestant nation, into outright enemies with Spain, a Catholic one. The English were motivated to undermine Spanish hegemony in the New World for both political and religious reasons.
In economic terms, the growth of the English population and its wars with other nations taxed national resources, particularly lumber and food. Colonization would not only address these issues, but strike at Spanish power as well. This led to the formation of the London Company (also known as the Virginia Company) which held a royal charter to establish and govern colonies in North America. It should be noted that virtually all other colonists, including the famed Puritans and Pilgrims fleeing religious persecution, operated under the domain of these royal charters as well. The ultimate goal of these charters was not to pursue human rights or alternative forms of government, but to turn a profit. Additionally, the religious persecutions in England frequently included economic restrictions, such as being forbidden to own land, hold public office, or teach. So, while certain colonies such as Massachusetts were strongly influenced by religious factors, economics was ultimately the reason for both the choice to leave England and the choice to colonize under her rule.