The Puritans and Quakers were similar in that both religious groups faced persecution in England and sought religious freedom in the American colonies; however, the religions differ widely in their beliefs. Quakerism, officially known as the Religious Society of Friends or simply Friends, arose in England as dissenters from the Church of England. Believers in the ability of people to follow what they called their "inner light" to reach a direct relationship with God without help from clergy, they were heavily persecuted in England until the late 1700s. As a result, many Quakers came to the New World, where they lived in the Massachusetts Bay Colony until many were banished. They also lived in other colonies, including Rhode Island and New Amsterdam (they also faced persecution in New Amsterdam). The Quaker leader William Penn founded the Pennsylvania colony in 1681 as a place where all religions, including Quakers, could practice freely. Quakers were opposed to slavery and pushed for its abolition. They also favored women's rights, pacifism, fair treatment for the mentally ill, and fair treatment of Native Americans.

The Puritans were similar to the Quakers in that they also faced persecution in England and went to the New World to find a place where they could practice their religion freely--first in the Massachusetts Bay Colony. They were followers of John Calvin, the Protestant leader who wanted to cleanse the Church of England of Roman Catholic rituals. While they, like the Quakers, were unhappy with the Church of England, they sought to reform it from within, and they remained part of that church. Unlike the Quakers, their government involved persecuting others and was not open to other religions. They also did not believe in the "inner light," or one's personal connection with God without intervention from clergy, and they did not believe in pacifism or the rights of Native Americans. Over time, many descendants of the original Puritans came to advocate abolitionism (the end of slavery), however. Puritans and Quakers were also similar in that they were known for their industriousness and hard work. 

Were there any similarities between the Puritans and Quakers? 

There was very little in the way of similarities between the Puritans and the Quakers in Colonial America.  In fact, the Puritans were sufficiently opposed to the Quakers that there were instances of Quaker missionaries being executed by Puritans.

The major reason for this was that the Quakers were much less dogmatic than the Puritans.  They believed that all people had the "Inner Light" of God within them and that listening to this "Inner Light" was much important than reading scripture or listening to what preachers had to say.  The Puritans wanted a rigidly controlled society in which all people worshipped God in the same way.  They felt that this was necessary to prevent God from punishing their community.  

The Puritans were much more controlling and dogmatic than the Quakers.  Therefore, they were very different in both religious doctrine and social habits.  The only real similarity is that both were Christian sects who went to the American colonies to try to have more of an ability to worship God in their own way.