Well, the answer is: not a lot. I agree with almost everything in it verbatim. It is a good, Calvinistic, Evangelical creed, but I thought it would be useful to mention the few areas where I disagree with it:
3. Of the going down of Christ into Hell.
As Christ died for us, and was buried, so also is it to be believed, that he went down into Hell.
I disagree with the mistaken interpretation of 1 Peter 3:19 that believes that Christ went down into Hell.
6. Of the Sufficiency of the Holy Scriptures for Salvation.
...And the other Books [the Apocrypha] (as Hierome saith) the Church doth read for example of life and instruction of manners; but yet doth it not apply them to establish any doctrine; such are these following...
This statement could be understood in the same sense as the better statement in the WCF (which came out of a revision of the 39 Articles), but the implication and the danger is that this statement means that the Apocrypha can be read in Church like Scripture, even though it is seen as inferior to "Canonical" Scripture. This is what has happened in practice and is a logical outworking of Anglicanism's semi-reformed view of sola Scriptura.
Moreover, the 39 Articles don't claim that the Holy Scriptures are the infallible and inerrant Word of God.
7. Of the Old Testament.
...the Law given from God by Moses... nor the Civil precepts thereof ought of necessity to be received in any commonwealth...
This is a more imbalanced (in my opinion) statement concerning the position of the Civil Law of Moses, compared to WCF 19:4: "To them also, as a body politic, he gave sundry judicial laws, which expired together with the state of that people, not obliging any other, now, further than the general equity thereof may require."
There is a general equity in the Civil Law that does make necessary requirements, and that equity is more than moral precepts, it is also equitable penology, for equity is about civil justice, not merely morality. Interestingly, the English Reformers Bishops Hooper and Latimer agree.
20. Of the Authority of the Church.
The Church hath power to decree Rites or Ceremonies... yet it is not lawful for the Church to ordain any thing that is contrary to God's Word written... it ought not to decree any thing against the same...
Here is the great difference between historical Presbyterianism/ Independancy (a.k.a. Puritanism/ Reformed) and Anglicanism: the Regulative Principle. We say you may only rule the Church as Scripture ordains, they say that you may create offices, etc., as you please. We say that you may only worship God as He has asked, they say that you may have any rites that you like as long as they are not forbidden. (Of course, most modern Presbyterian and Baptists are really Anglicans/Lutherans and they don't even know it!)
21. Of the Authority of General Councils.
General Councils may not be gathered together without the commandment and will of Princes...
This is Erastianism. In opposition to this, the Puritans stated in WCF 31:2, "...the ministers of Christ, of themselves, by virtue of their office, or they, with other fit persons, upon delegation from their churches, may meet together in such assemblies."
34. Of the Traditions of the Church.
It is not necessary that Traditions and Ceremonies be in all places one, or utterly like; for at all times they have been divers, and may be changed according to the diversity of countries, times, and men's manners, so that nothing be ordained against God's Word...
This repeats the error of Article 20 contra the Regulative Principle. Of course, this can be given a Reformed interpretation because it is against God's Word to add to or take from what He has ordained. However, this is not how it is interpreted, so we must understand it with the intent of the framers.
36. Of Consecration of Bishops and Ministers.
The Book of Consecration of Bishops, and Ordering of Priests and Deacons... whosoever are consecrated or ordered according to the Rites of that Book... we decree all such to be rightly, orderly, and lawfully consecrated and ordered.
Bishops, priests and deacons, as conceived by the Anglican Church are unbiblical offices. Bishops/ overseers are the same as elders/ presbyters and there is parity between all elders, and deacons have a temporal caring and administrative role, not a ruling and teaching ministry.
37. Of the Power of the Civil Magistrates.
The King's Majesty hath the chief power in this Realm of England, and other his Dominions, unto whom the chief Government of all Estates of this Realm, whether they be Ecclesiastical or Civil...
The 1801 CoE amendment rightly removes the power of the Civil Magistrate over the Church, but the CoI still has the original 1571 rendering as above. Again, this is Erastian.
By the way, what every Irish Anglican should read is the far better Irish Articles of 1615, which were the confession of Irish Anglicanism, before the English Church imposed its will, and which had a profound influence on the WCF. (Good ol' Ussher! Pity he allowed his Erastianism to get in the way!)
