@Ryan&Amber2013: As I've said on another thread, I'm not going to get into this in any depth without clearance from the site administrators since it will necessarily involve criticism of Presbyterian polity.
@NaphtaliPress is here, and has given you a reference to one of the key documents during the Westminster era of debate between the Congregationalists at Westminster (known as the "Dissenting Brethren") and the Presbyterians.
He is correct about the problems of reading that debate. The Congregationalists didn't yet have a formal statement of polity at the time.
That changed later.
An important caution applies here: Confessions count. We expect that when Arminians or Baptists criticize us, that they won't go to some random Calvinist who may or may not represent the Reformed faith, but rather go to one or more of the Reformed confessions. I don't blame Presbyterians for the PC(USA) abuses, and I know Chris from his writings well enough to know that he cares about accurately representing Baptists, Congregationalists, and others, even when he disagrees.
It's also important to remember that the Dissenting Brethren, while they dissented, were BRETHREN. Both sides of the Presbyterian-Congregational debate were at Westminster because it was understood that they shared the same faith but not the same polity. We need to regard each other as brothers in the Reformed faith, and treat each other in a brotherly manner.
So let's go to the confessions.
There are differences between English Congregationalism and New England Congregationalism. The main difference is that the New Englanders benefitted from actual practical experience in governing their churches without a presbytery or a bishop. The Dissenting Brethren at Westminster knew what the Puritans in New England and the Dutch were doing -- as
@De Jager points out, Dutch Reformed polity predates Westminster and has aspects of both systems of government, and in its modern form, has largely been defined by Abraham Kuyper who was (I believe correctly) regarded as a "crypto-Congregationalist" -- but the English Congregationalists didn't have a lot of practical experience. That was a significant part of why the New England Congregationalists called the Cambridge Synod, which approved the Westminster Confession with certain stated exceptions that are generally what one would expect from a Congregational synod, and in 1648 issued the much more detailed Cambridge Platform which functioned as the "church order" for New England Congregationalism.
Political and ecclesiastical reality meant that in England, the Congregationalists waited to call their own synod, which in 1658 revised the Westminster Confession to produce the Savoy Declaration of Faith and Order. That "and Order" is important. The "order" is appended to the Savoy Declaration. It is a much less detailed form of church government that doesn't adopt the Cambridge Platform in detail and leaves a lot of openness to different views of how local churches should be governed and how they should relate to each other.
Of course, the overthrow of Cromwell's son just a few years later meant that the Savoy Declaration of Faith and Order became something of a "dead letter." Congregationalists and Presbyterians were both thrown into persecution, and the English Congregationalists did not get the opportunity to "flesh out" their views on church government.
The result is that the New Englanders were left with a considerably more detailed "jure divino" statement of how their churches should be governed, with detailed Scripture proofs, parallel to what the Presbyterians had, while the English Congregationalists had a lot looser arrangement with a lot more diversity. That situation continued even after persecution ended and is an ongoing difference between the English and American traditions of Congregationalism, with the British (yes, there are small numbers of Congregationalists in Scotland and Ulster, not just England) using the Savoy Declaration of Faith and Order as their confession, and the Americans using two confessional documents, the Savoy Declaration and the Cambridge Platform. (I'm simplifying here a little -- there were some other American confessional statements that were written and had some regional usage, but the Savoy was widely used as the "Congregational version" of the Westminster Confession, and the Westminster Shorter Catechism, while I'm aware of only one case in which it was formally adopted by a regional assembly as a formal doctrinal standard, was widely used for catechesis. Today, I think the Heidelberg Catechism is more commonly used for local catechesis, and the church of which I was a member for many years had formally adopted it, along with the Savoy and Cambridge, as its doctrinal standards.)
A difference on the two sides of the Atlantic is not unique to Congregationalism. We see somewhat similar issues of difference in how British Presbyterianism (particularly Scotland and Ulster) works, compared to American Presbyterianism, though the reasons are different.
That is a basic introduction to the history.
Here are the key issues on how the three systems work out:
1. Both Presbyterians and Congregationalists agree -- and on this point, they both differ from the Dutch Reformed -- that not just basic principles of church government but specifics can be found from Scripture.
2. The Dutch confessions make certain things mandatory about church government, but the federations are viewed as being for the well-being (bene esse) of the church, not for their very being (esse), and that means there are things in the Church Order of Dordt that make no claim to have explicit biblical warrant. That's not a criticism of the Dutch, with whom I largely agree on polity (at least post-Kuyper), but simply a recognition that their confessions are of an earlier date before some of the later church polity differences got "hashed out."
3. Historic confessional Presbyterians and Congregationalists will both agree that if something can't be found in the Bible, it shouldn't be found in the church. The Dutch may think that's strange, but it's simply applying the regulative principle of worship to the governance of the church. Has God given a government to His church, or do we have freedom to order the church according to practical considerations and preferences? Both the Presbyterians and the Congregationalists agreed that we need to go to the Bible for warrant for what we do.
4. Where that gets difficult is determining WHAT the Bible says on specific issues. The Cambridge Platform makes the case that there is no such thing as a "regional church" with officers over it, and the only biblical officers of the church are elders and deacons, and that after the age of the apostles, no elder has authority beyond his own local church, though churches can, should, and sometimes MUST seek counsel of other churches to resolve their problems. Presbyterians make a quite different case for the existence of regional churches.
5. That difference comes out in important ways in the three ecclesiastical systems. A Dutch pastor or a Congregational pastor is a member of a local church and accountable to his local elders. A Presbyterian pastor is accountable to his presbytery. That is most emphatically NOT a minor matter. A Dutch conservative today is going to say that if a local church objects to a decision of the synod, the synod has NO AUTHORITY to depose the pastor or the elders, and the most it can do is set the church outside the federation. That is not the Presbyterian view of the authority of higher courts.
I can get into a lot of detail here, and the Cambridge Platform does that. So does the Westminster Form of Presbyterial Church Government.
The key question is -- or at least should be -- first, what God has or had not said, and second, whether we can do things in the church that God has not told us to do.
How we answer those questions will determine where we end up on church government.
I sincerely hope I have been respectful, and spoken in a brotherly manner, to those with whom I dissent, and about the points on which I dissent.
I owe that to brothers in the Lord, especially those who affirm sovereign grace.
There just aren't enough Reformed people out there to be fighting bitterly over these matters. Disagreement is unavoidable. Bitterness is to be avoided.