Historians have long identified the cultural changes in the Enlightenment period as the precursor of the Atlantic Revolutions, but others are now saying that the social changes in the West, including the rise of the urban sector, also helped brew the Atlantic Revolutions. Which had a greater impact?

It's not really possible to measure which had the greatest impact. One can't do controlled experiments, taking out one variable to see how the other works. Instead, it is more likely that several different factors worked synergistically, mutually reinforcing one another.
Max Weber seminally argued that one of the causes of the rise of modern industrial capitalism was the growth of Protestantism. It is possible to look at how the growth of Enlightenment ideals, urbanization, and religion worked together.
The first issue is that Protestantism meant a break from the universal Roman Catholicism of the medieval period and a movement toward the notion of free choice in religion. Often this took a nationalist bent, with religious identity tied to the rise of nationalism. The United States, for example, saw itself as breaking away from England, which had an official state church; many of the original immigrants were dissenters (people not members of the Church of England) seeking freedom of religion. Irish Roman Catholics resented subjugation to English Protestants. In France, the alliance of "throne and altar" and the restriction of highly paid positions in the church, supported by tithing, to the nobility, were among the grievances leading to the French Revolution. Thus Enlightenment ideals of religious freedom contributed directly to the Atlantic Revolutions.
Urbanism fits with this in several ways. Many of the people most open to Enlightenment ideals were part of the growing urban bourgeoisie. They were neither poor peasants nor part of the hereditary aristocracy, and they actively demanded a share in power. As a group, they were well educated and often met in urban coffee houses, dissenting churches, dissenting academies (schools not affiliated with official churches), and various clubs and associations where they could exchange ideas and coordinate activities concerning social and political change. Economic change and urbanization thus provided the enabling conditions for circulation of Enlightenment ideas and also a class of educated, financially comfortable people who resented being left out of power structures based on heredity and religious conformity.