I like this question because it forces us to read the Westminster Confession in its context, and not just as a settled statement of belief. In summary, the two confessions are in continuity, but the earlier Scots Confession was more permissive. Especially, it did not exclude the option of episcopal polity, or of royal power over the Church. The Westminster Confession includes more ramified theological content, and is more strongly Calvinist. This is partly the result of its having been written by theological experts, and vetted by others in the Scottish universities before its adoption, and partly because the writers were commissioned to make sure there would be an official "weapon" to use against Arminians and so on.
The project of the assembly at Westminster was to create a unified religious settlement for the three kingdoms (England, Scotland, Ireland), incorporating reformed doctrine, polity, procedure for worship, psalm and prayer texts, and relationship with the civil order. It was the product of a "covenant" between presbyterians in the different nations, seeking to establish their ideas over Catholicism on the one hand, and more radical Protestant factions on the other. In the period, the kingdoms were engaged in civil war, and this project was a claim for how it should be resolved.
From official correspondence from the Scottish side, the main doctrinal points they thought should be included in the settlement were:
- Conformity in the three kingdoms. Everyone would have to follow the new Confession and related documents. They also envisaged that this would be an example for the world, both morally (look how holy we are) and practically (we've written this stuff for you already, just sign up).
- Presbyterian polity. This was opposed to royalist counterproposals for retaining bishops in some capacity or in some areas, and to more radical congregational views that would have eliminated any church structure beyond the parish.
- Excommunication. Ministers - and nobody else - were to have authority to exclude someone from the sacraments. In particular, they could rebuke even the King, and on the other hand the King would not have any authority over the administration of the sacraments. They were also keen on a general suppression of vice and immorality (drunkenness, non-observance of the Sabbath, etc.).
- Elimination of heresy. "Socinianisme, Arminianisme, Anabaptisme, Antinomianisme, Brownisme, Erastianisme, Independency, and that which is called (by abuse of the word) Liberty of Conscience, being indeed liberty of errour, scandall, schisme, heresie, dishonouring God, opposing the truth, hindering reformation, and seducing others; whereunto we adde those Nullifidians, or men of no religion, commonly called Seekers" were to be suppressed, by power of the civil magistrate and with the authority of doctrinal statements against them. 1
Comparison with the Scots Confession shows that the Westminster document is more explicit about doctrine, including Calvinist theories of soteriology rather than more general statements that might cover Arminian views as well. It also declares a stricter separation between the roles of Crown and Kirk. Under Westminster, regulation of the ministry of word and sacrament is excluded from the responsibility of the civil magistrate: he must suppress heresy and keep order, but it's for the Kirk to decide what heresy is. The Scots Confession leaves it open that the King's role for "maintenance of the true religion" could be more expansive, perhaps including the power to excommunicate or readmit.
The practical difference between the confessions was shown in the Restoration. On 28 March 1661, Charles II revoked the Westminster Confession via his "Act Rescissory", which repealed all legislation from 1633 onwards. He declared that the Protestant religion would be observed as it had been under James VI and Charles I; and while he said Presbyterian polity would be maintained "in the mean Time" if presbyteries were "keeping within Bounds and behaving themselves", it was only a few months before episcopacy was brought back.2 In effect, the Scots Confession had been restored as the standard of faith, and read in a way that allowed considerable royal latitude, as well as the presence of bishops.
In 1681, the Test Act3 formalized this, at the suggestion of James Dalrymple: its loyalty oath invoked the Confession "recorded in the first Parliament of king James VI", and required the belief in royal supremacy in all matters, "as well ecclesiastical as civil". Religious assemblies were only allowed if the king approved them, and they certainly could not "endeavour any change or alteration in the government, either in church or state". While this was sold as an anti-Catholic measure, it was really an attempt to wrest Protestants back to a more moderate course. (And James VII exempted Catholic officers from having to follow it anyway!) Accordingly, the re-ratification of Westminster in 1690 made it absolutely clear that the presbyterian system would be restored, including that it would be "the only government of Christs Church within this Kingdome". The royal power was taken away, and this settlement was preserved in subsequent legislation, including the Act of Union with England. To this day, doctrine and governance of the Church of Scotland is for the Church itself, and not for the Queen or Parliament.
1. Declaration and Brotherly Exhortation of the General Assembly of the Church of Scotland to their Brethren of England, 20 August 1647.
2. The King's Majesty's Proclamation concerning Church-affairs, 10 June 1661.
3. An Act anent Religion and the Test, 31 August 1681