It appears to me that the first paragraph was simply meant to set up the second. That is because there can be no difference between the Continuing Churches that practice the traditional Catholic faith and practice of Anglican Christianity, whether the worship is decidedly high, mid, or low. Most of us like the high church sort of worship, but I trust we know better than to think that incense, chasubles, benedictions, holy water fonts or even Marian devotions are the essence of the Catholic faith (although the last, properly understood, is centered on the Incarnation, and is in that theological sense tied to the essence). The celebration of a low church Prayer Book Holy Communion is just as valid as a high church Mass from the Anglican Missal. Sometimes, that low celebration is refreshing in its simplicity, and helps us feel the depth of our faith for that reason, just as a very high Mass helps us to transcend the things of earth. As a lifelong Anglican I embrace all of these expressions of the Catholic faith in celebrating the Eucharist.
The real question in the first paragraph deals not with these different ways to worship, but with the substance of doctrine. We cannot answer what the difference is, because another question would get a simple answer of "no." That question, which the correspondent did not ask, is, "Is there a difference between the teaching of the Catholic Church and the teaching of the Continuing Churches?" The answer is no.
"And the Catholic Faith is this: That we worship one God in Trinity, and Trinity in Unity, neither confounding the Persons, nor dividing the Substance." -the Athanasian Creed (Quicumque vult)
The real question is why we call the Roman Catholic Church by that name. First of all, it is not a sign of disrespect, since we regard the Church of Rome and those churches in communion with it, to be part of one and the same Body of Christ to which we ourselves also belong. Also, it is a proper name of the largest Rite in that Communion, namely the Latin Rite.
We must refer to the Roman Catholic Church by that name, and this is for the very reason mentioned in the question. When we speak of "the Holy Catholic Church"and the "One, Holy, Catholic and Apostolic Church," we are not professing belief in some foreign institution that we merely aspire to, but the Church in which we live as members of the Body of Christ. The Orthodox Church has the same problem with the name of the Roman Communion; they too see themselves as within the Catholic Church of the Creeds.
It is more basic than the fact that we have retained a valid Apostolic Succession of bishops. It is rooted in baptism and in faith in Christ, and a continuation of the teaching handed down from the beginning. Whereas we mean no offense to the churches in communion with Rome, neither do we offer an apology for recognizing our own validity and taking joy in our true identity.