"Prayer Book Catholic" is not a term one hears very often these days. I have a suspicion that that may have something to do firstly with the disappearence of the Prayer Book in England, and partly because the liturgical revision has led to a drawing together of the mainstream A-C and Prayer Book strands in the Church of England. However, the term still has some relevance when talking about Anglo-Catholicism in the Continuum, even though "Prayer Book Catholic" has been a term little used in the USA.
One great characteristic of the American Continuum is that it is overwhelmingly a "small-c catholic" endeavour. Even the "Low Churchmen" can tick all the basic Anglo-catholic boxes - belief in the authority of Scripture, in the three Creeds, in the importance of the Early Fathers and Councils, in the apostolic origin of Holy Orders, in the Real Presence, etc.. The areas of dispute tend to lie in areas that are not core doctrine, such as what should we believe about the Blessed Virgin Mary and how important are Counter-Reformation devotions such as Benediction, the Sacred Heart of Jesus and the Immaculate Heart of Mary. Thus, within the Continuing environment "Prayer Book Catholicism" represents a sort of middle ground, firmly rooted in the Prayer Book tradition of worship on the one hand, and firmly catholic in doctrine on the other.
Prayer Book Catholicism seems to have had its origin in the Bisley School of Tractarianism, and among those Ritualists who thought that being Anglican was at least as important as being Catholic. Doctrinally, it tended to have a more of an interest in the Early Fathers than it did in modern Roman Catholic theology and placed a strong emphasis on Scripture as the test of what should be dogma. This basic theologial emphasis will give you an idea of where and how moderate Anglo-Catholic theologians differed from Rome. The obvious areas of dispute are those dogmas, doctrines and disciplines of the Roman Church that deviate from the practice of the first eight Christian Centuries. This resulted in a strongly Biblical expression of Catholicism that, in practical terms, valued the integration the sacramental Christianity into the personal commitment to Christ and a striving after holiness characteristic of the old Evangelicalism.
In ceremonial terms Prayer Book Catholicism developed along two paths. The slightly older appraoch is the "English Use" which is a deliberate attempt to revive the such late mediaeval ceremonies as are necessary to the Prayer Book liturgy. This approach was first explored in the mid-nineteenth century, then again c. 1900. It was always most popular in the UK, but, apart from a brief vogue in the 1920s and again in liberal Catholic circles c. 1950, the English Use never really caught on in the USA.
The characteristic Anglo-Catholic approach to ceremonial in the USA, where there is no Ornaments Rubric to point us to the "second year of King Edward the Sixth" has been what I call either "Fortescue and Water" or "Ritual Notes and Water." This approach, adopting and adapting the Tridentine Roman ceremonial in celebrating the Prayer Book services, dates from around 1875 and became popular due to the availability of mass produced items for this style of liturgy. Strangely this led to one of the more visible characteristics of American "Prayer Book Catholicism" - when Roman Usage and the BCP came into conflict, the BCP usually won. That said, there are some widespread additions to the BCP Communion service of which Prayer Catholics are guilty. The most popular of these have been the use of some of the personal prayers of the celebrant, and the addition of the Benedictus, and the Agnus Dei to the Communion office, and less frequently, the use of the Ecce, Agnus Dei before Communion. Also some of the commoner visual elements peculiar to the older Roman Rite - "the big six" on the altar, tabernacles, genuflection, the major elevations, cassocks and cottas for servers have been widely adopted.
Some folks have been tempted to label this Romanizing, but I think it represents the evolution of an American version of Catholic Anglicanism. It is also so firmly entrenched that to launch a new series of liturgy wars aimed at making everyone English Use would be a sinful waste of time and energy when there are souls perishing for want of the catholic Gospel of Christ. There is also a level on which it seems a little odd to tie a country and culture that evolved Post-Reformation to a mediaeval aesthetic. On the other, we should not allow our adoption and adaption the Baroque liturgical aesthetic become a simple ape-ing of 1750s, 1850s or 1950s Rome.
At the end of the day, what Prayer Book Catholic need to be careful to do is be "Bible" or "Evangelical" Catholics. This approach requires our expression of Catholic theology to be rooted in the Scriptures (as expounded in the Early Fathers) and for our worship to be sacramental, dignified, beautiful. Concern for the outward in worship is an extension of the incarnational principle in Christian theology. God-made-man teaches us that the created world special to God and also reminds us that man is made in God's image. It also says something very important about the sacredness of the physical world which was, after all, created by God. This leads us to conclude that the use of human talent, beautiful art, vestments and music, and the engagement of the senses in worship all reflect the incarnational approach to Faith and are an offering acceptable to God. The incipient puritanism of much of American Protestant worship reflects as sort of muted dualism where the physical is "bad" whilst the spiritual is the only true good. This can lead to other disconnects - not just between faith and art, but between religion and morality, faith and conduct, etc.. The common repudiation of art, good music, ceremonial and ritual in worship reflect a religion that over emphasizes the Atonement at the expense of the Incarnation, or more dangerously, especially in the liberal circles, places too much emphasis on Jesus the Great Exemplar and Social Reformer, and not enough of Jesus God Incarnate, our Redeemer, the Christ.
As Christianity becomes marginalized by the Secular Progressive movement (and if you do not believe that is happening look at Western Europe) it becomes increasingly important to take a holistic approach to religion. Christianity should affect every part of the believer's life, not just Sunday mornings and Holydays of Obligation. Christianity should be taught not just as a religion, but as a way of life, a culture, an integrated system. Detaching Christianity from life - which is the logical consequence of teaching it as "just a religion" - actually plays into the hands of the secular progressives as it makes it far easier for them to portray Christianity as irrelevant to daily life - pie in the sky when you die.
When I talk about holistic religion, that naturally brings to mind another word that derives from the same Greek root - catholic. The essence of the catholic faith is both redemptive and incarnation. It values the physical as well as the spiritual as God-made. It is a culture as well as a religion. Thus in our teaching of the faith we need to teach ot just Christian theology, but Christian morality, Christian Liturgy, Christian Art, Christian Culture. If the Twenty-first Century is going to be the twenty-first Christian Century the Church is going to have to teach the fullness of Christianity in order to resist the in-roads of the Truth's two great enemies Secularism, and Islam. They are not shy about (mis)representing their errors as integrated world view; and we should be forceful in our declaration of Christian as the way, the truth, and the life.