Both the paleocon and neocon wings of the Catholic Right view faith as an essential element for a viable society.  It is, however how faith is to be put to work that separates the two philosophies.

Both camps desire a traditional Church, free of dissent.  One step in that direction has been to make the Church uncomfortable for non-traditionalists and current and would be dissenters from their agenda - to make the most well-organized religious institution in the world a major force for political reaction.

Arguably the paleocon vision is the less nefarious.  One of the longstanding beliefs of more traditional conservatism is that American culture is intertwined with a Judeo-Christian heritage.  It's politics are informed by the Enlightenment-derived notion that puts its faith into the judgment of the citizenry - provided, of course that that same citizenry maintains its "traditional" religious-based values such as opposition to abortion, LGBT rights and in the Catholic Right's case, birth control and embryonic stem cell research.  Hence, the paleocon desire to make a traditionalist, pre-Vatican II Catholic Church is predicated upon a populist approach to government, one in which a populace is virtuous by adherence to their version of Catholic orthodoxy.

They further believe that the Church can be trusted as an arbiter of international disputes - at least to the point where the United States could be kept out of international conflicts.  That was, for example, the paleocon impulse of seventy years ago to avoid fighting Hitler, defining (albeit, in that scenario, incorrect) as a needless foreign entanglement.  This view was echoed in their opposition to the 2003 invasion of Iraq.  Implicit in the isolationism of both 1940 and 2003 was the desire for Americans to be left alone in peace.  Taking that though one step further, more traditional conservatives generally desire a world where "swords are beaten into ploughshares" --provided a socially conservative America could maintain its security.

The Catholic neocon view is quite different with regard to both these issues.

Neocons, Catholic or otherwise, do not much place stock in the possibility of a virtuous citizenry.  Instead, they rely on an educated elite whose job it is to lead and control the masses.

Recently I examined the differences between the paleocon and neocon Catholic Right.  Among those differences:  "Where the Neos see faith as tool for empire, the Paleos see it more as a bulwark against change:  Tradition for tradition's sake and nothing more.  To their credit, the Paleos often call the Neos on this cynical use of faith."

But here is the rub:  neocons, drawing upon the writings of philosopher Leo Strauss cannot imagine a world built upon a universal vision of world peace - a clear mission of the Catholic Church.  Indeed, Strauss himself wrote in his infamous essay Jerusalem and Athens, "The cessation of evils that Socrates expects from the establishment of the best regime will not include the cessation of war."

Strauss's apparent disdain for a cessation of international violence has not been lost upon neoconservative Catholicism's greatest provocateur, George Weigel.  Ever since his Reagan-era involvement with supporting the Contras in Nicaragua, Weigel - whenever given the choice between restraint and militaristic policy pursuits - consistently opts for the latter.  Consider this synopsis from Right Web:

Long before 9/11 and the Iraq War, Weigel had been calling for a more aggressive post-Cold War U.S. foreign policy. In a 1995 special issue of the Hoover Institution's Policy Review, Weigel argued that the United States "has not had a foreign policy since January 20, 1993. Before then, with the exception of the Gulf War, the foreign and defense policies of the [George H.W. Bush] administration were geared not to shaping the post-Cold War future, but to managing the Cold War's endgame. The Republican president inaugurated on January 20, 1997, will thus have an immense responsibility: creating the first post-Cold War foreign policy worthy of the name" (Policy Review, Fall 1995).  According to Weigel, such a policy must include the aggressive expansion of NATO; the development of a policy of "preemptive military action [to] be used to counter weapons proliferation against rogue states or terrorist organizations"; developing and sharing with friends missile defense systems as an "essential technological complement to an assertive policy of nonproliferation and counterterrorism"; aiding the "democratic opposition" in countries across the globe, including in China and Cuba; and reforming the United Nations, which "has become a hotbed of internationalized libertinism, as demonstrated by the 1994 (anti-) population conference in Cairo and the 1995 Beijing conference on women."

As well as:

Weigel endeavored to develop a Christian justification for the invasion of Iraq and for the use of preemptive military force. In opposition to the arguments of many leading Catholics, Weigel stated that the Catholic just-war tradition "lives more vigorously ... at the higher levels of the Pentagon than ... in certain offices at the United States Conference of Catholic Bishops." In response, the National Catholic Reporter editorialized, "It's an interesting argument, but to employ a military euphemism, Weigel seems guilty of faulty targeting. The U.S. bishops have put out one well-reasoned, cautious statement expressing reservations about a possible attack in Iraq, but there has been no antiwar campaign from their headquarters in Washington. The real outcry in the Catholic world is coming from across the Atlantic Ocean, and more precisely from the subject of Weigel's 1999 biography Witness to Hope-Pope John Paul II. If Weigel should be picking on anyone, it's the pope" (National Catholic Reporter, January 2003).

Catholic neocons see in the Church an organizational structire that makes her so politically desirable for empire-building. Damon Linker, on pages 68-69 of his revealing work, The Theocons:  Secular America Under Siege described it through the eyes of the late Richard John Neuhaus:

"...Neuhaus believed that the promise of uniting the theoconservative movement with the Catholic Church was so great that the effort had to be attempted. Catholics were, first of all, the single largest religious group in the country, making it exceedingly difficult if not impossible to launch a successful program for political and religious reform in the country without significant support from within the ranks of the Catholic faithful.  Then there was the church's long history of theological and political reflection, which made Catholics far more competent than evangelicals and other Protestants to take the lead in pressing religiously based moral arguments in the nation's political life."

A few passages further on, Linker hits the nail on the head:

But most promising of all was the Vatican's robust defense of ecclesiastical authority.  Unlike the Protestant mainline, whose leadership had come to preach unorthodox, antitraditionalist views, the heads of the Catholic Church in Rome (Pope John Paul II and Cardinal Ratzinger) refused to compromise with or capitulate to blatant theological deviancy.

And then:

Gazing across the Atlantic to St. Peter's Basilica, then, Neuhaus believed he spied enormously powerful allies in his struggle against secularism in the modern world -- Christian intellectuals who had undertaken their own theoconservative project within the church who longed to engage the enemies of Christian truth wherever they were found, including in the political realm.

What neos such as Weigel, Michael Novak and Neuhaus, esoterically advocate is not a Church that sets a moral teaching that is to be followed by all, but one where neoconservative elites such as themselves advise and prod the Vatican towards an agenda more akin to neoconservatism than to Jesus Christ.  Essentially, they seek a "Church within a Church," a small circle of elites guiding it away from either the pacifism of the left or the isolationism of the traditional-minded paleos.

If you think this seems implausible, consider neoconservative movement founder Irving Kristol's take on Christianity.

Writing in chapter 37 of his tome, Neoconservatism:  The Autobiography of an Idea, Kristol decries Christianity as countercultural, conflating its traditional deference to the poor with Marxism all while not-so-esoterically complaining that Vatican II was Catholicism's capitulation to socialism and modernity (the chapter was a reprint of a 1979 lecture given shortly after the ascendancy of Pop John Paul II).  He praise Pope Pius IX 's strident anti-modern, anti-democratic screed Syllabus of Errors. In a clarion call for a return to a reign of dissent-free orthodoxy, Kristol then lectures the Church:

Go tell the young people that the message of the [Catholic] church is to wear sackcloth and ashes and to walk on nails to Rome, and they would do it.  The church turned the wrong way.  It went to modernity at the very moment when modernity was being challenged, when the secular gnostic impulse was already in the process of dissolution.

The Catholic Right, particularly Church neocons have taken Irving Kristol's call to leap backwards to heart, so much so that they are more than willing to tear the Church apart.

But as of late this neoconservative infiltration of the Vatican has been somewhat derailed.  Weigel and Novak who once had close ties to the late John Paul II now find themselves a step or two removed from Pope Benedict XVI's inner circle (Weigel was John Paul II chosen biographer while Novak had the late pontiff's ear on economics).  Much to the neos' collective horror the current pope seems to be embracing the paleos' view on biological issues while returning to historic Catholic approaches to social justice on economics  This was on full display in the recent encyclical, Caritas in veritate ("Charity in truth").

In any event, the desire for constructive schism is still present among many Church conservatives who while disagreeing on the ends, are certainly united in their desire to get rid of moderates and liberals. That much is certain.