Religion in the United States, at least, is so far from what it was back then that it's like they're hardly the same thing.
In Puritan areas of this time, the Church and the government were one and the same. People were subject to intense peer pressure to behave properly, but it went way beyond that. People could actually be punished by the government for "wrong" religious beliefs and actions. For example, Quaker missionaries to the Puritan colonies were sometimes (not often, but sometimes) executed. In addition, people who had the "wrong" beliefs, like Roger Williams, could be thrown out of the colonies.
Nowadays, we worry about excessive government promotion of religion when people say the Pledge of Allegiance or when there's a Christmas tree somewhere. Back then, religion was enforced by the government as well as by strong social pressures which are not present in very many communities in the United Sates today.
When it comes to theology, religion was much more strict back then. People were expected to believe in an angry and harsh (but just) God. This is very much in contrast to today's more "feel-good" religious beliefs -- the emphasis on a loving God.
This is a rather broad question that can be taken in different ways. Part of the reason for this is because religion is not a monolithic thing. There is great variability both then and now. The Puritans at their heyday were Calvinistic in their theology with a strong sense of sin, a strong work ethic and had a desire to be an example of God's grace to the world. If you think about it, there are many groups today that believe in the same thing. Any conservative presbyterian would pretty much agree with Puritanism of the past. However, things have also changed. Many people are no longer devout when it comes to religion. Even if they believe in God, many people are pretty lax when it comes to theology and practice. So, in short, the answer is very variable.
This is quite a broad question. I think that one primary difference between the religion during the Puritan times and now is the heterogeneous nature of the American landscape. At the time of the Puritans, there was essentially one notion of worship, one dogma of religious experience that all had to follow. Essentially, the Puritans were "the only game in town." Their strict codes of conduct had to be adhered with the most stringent of standards and interpretation. This is not the case now as there are many different forms of worship that are not only limited to Christianity. The fact that there is a multiplicity of paths to spiritual conception as well as the fact that our social order has become more permissive and tolerant in the nature of religion might make another level of difference between both settings. There are still dogmatic orders, but I would also submit that these are not the prevailing majority in America as the Puritans were during their time frame.
In the Bible Belt where there is Fundamentalism one may find many parallels between the Puritans of old since the churches have been perpetuated from the Puritan ones. For instance, the Fundamentalists believe like the Puritans that God has a direct hand in their daily life. When things happen, they "happen for a reason," or "they are all part of God's plan." There is the Calvinist sense of predestination still in the Bible Belt as well as the belief in people's being "saved." Many of the Baptists in the South believe that sinners should be shunned, much like the Puritan villagers do in The Scarlet Letter. Like the Calvinists, some also believe that only a select number will enter heaven, and like the Puritan/Calvinist belief, Southern Baptists believe that the Bible is the ultimate authority.
The answer to the question certainly depends upon what area of the country one considers. Wherever the Religious Right is--Texas, many of the Southern states--beliefs are much more Puritanical than in the Midwest or Northeast, for instance. Then, in states such as California and Florida which have so very many modern immigrants, the religious beliefs/ideologies are much different from other parts of the U.S.
The Puritans had simple religious beliefs and a simple manner of worship. Most of the Puritans wanted to get rid of the costly, ornamental vestiges that cluttered other religions, namely the robes, ceremonies, and icons. Some wanted to get rid of stained glass windows, statues, and religious music. They wore simple, plain, drab clothing, attended church on the Sabbath in a plain wooden building, and strictly held to the teachings in the Bible. They were mostly quiet, reserved, and usually not in the limelight of politics, industry, or manufacture. They did not care for show, notoriety, or fame.
In contrast, the religions of today are often associated with stylish clothing, upper-class wealthy people, and fancy cars to brandish their adherrance to God. There are often pledge drives and religious services broadcast on television and radio to recruit followers. They build immense, lavish churches, and remind me of the words in the Scriptures: "They draw near to me with their lips, but their hearts are far from me."
Of course, not all religions are like this as there are certainly good people in all sects who try to live their religion devotedly and quietly without all the hype.