When the Magisterium ceased to say that the Catholic Church simply "is" the one true Church of Jesus Christ, and began saying that the one true Church of Christ "subsists in" the Catholic Church, the aim was not, as many trads charged, to negate the older teaching but to clarify the the status of non-Catholic ecclesial bodies. For the fact is that there are countless millions of people who belong to Christ by a form of baptism always recognized as valid by the Catholic Church, but who are not Catholics; thus as individuals, they enjoy an "imperfect" communion with Christ and the Catholic Church; but until Vatican II, there wasn't any clear and widely disseminated answer to the question how the ecclesial bodies to which they belong relate severally to the Catholic Church. Hence Vatican II's and the CDF's use of term 'subsists'.While the notion of "subsistence" is clearly a soundly Thomistic one, and represents a genuine advancement in sophistication over the older identity relation expressed by the very "to be", there is a down-side to this particular development of doctrine.
That term comes from the same Latin root as the noun "substance." In Catholic theology, a substance is ordinarily understood to be a unitary whole of a certain kind that perdures, and thus "subsists," through various activities and changes, which can include the sort of damage that consists in the loss of certain parts. Every human person, e.g., is a substance in that sense; one's bodily organs and cells are only parts that can remain alive (at least for a time, by nature or by artifice) while detached from the whole, but which have their full and proper reality only as parts of the living substance that is the person. Now the one true Church of Christ, as is clear from both Scripture and Tradition, is the universal "body of Christ" and thus, by analogy, a substance in the above-defined sense. Her high-level constituents are like organs or limbs: local churches, as the term 'church' is defined in the CDF document. Her base-level constituents are like cells: those individuals who belong to Christ and the Church by valid baptism but who might or might not belong either to some true, particular church or to an "ecclesial community" that doesn't quite qualify as a church. To say, therefore, that the one true Church of Christ "subsists" in the Catholic Church is to say that the Catholic Church is where the one true Church of Christ exists as a perduring whole, containing all the parts necessary thereto. To refuse to speak thus of other true but particular churches, such as the Orthodox, is logically equivalent to saying that they are properly parts of that subsistent whole which is the Catholic Church but exist to some extent apart from that whole, which is detrimental to both the whole and the parts themselves. It is detrimental to the whole inasmuch the whole, while still functioning as that integral whole which is "the" Church, does not fully embrace some of her proper parts. The whole remains what she is, but wounded. It is detrimental to the parts inasmuch as, while still being true churches and means of sanctification, they are not fully integrated into that subsistent whole of which they are proper parts, and thus can no longer manifest Catholic unity.
Whenever we speak of a unitary whole that is nonetheless by its very nature composed of parts, we raise the question of the constitutive relation between the whole and the parts. This is a question from an age old area of metaphysics known as mereology, the philosophy of the relations between parts and wholes. Let me illustrate one of the central issues at stake by means of a rather humble example. Consider two things, which we may for our purposes classify as entities, both of which contain parts: a loaf of bread and a television set. The TV clearly consists of parts: it is built out of wires, capacitors, transistors, and things of that sort. The loaf of bread also consists of parts, but their status qua parts is not as immediately evident: it is made out of flour, yeast, and water, and perhaps a few other ingredients depending on what kind of bread it is. Here is one difference between the TV and the loaf of bread: I can disassemble the TV and reassemble it as many times as I like. It is relatively easy to do, provided one knows how to do such things. I cannot, however, "disassemble" the loaf of bread and find, spread out on my kitchen table, a pile of fresh flour, some yeast, and a cup of water. Of both entities, however, we may say something like the following: "A TV is a set of wires, capacitors, transistors, etc.", and "A loaf of bread is flour, yeast, and water."
We may also say that a TV subsists in this particular collection of wires, capacitors, transistors, etc. That is, suppose one of the capacitors wears out, causing the TV to cease functioning as a TV. I can replace that capacitor, restoring functionality to the TV, and we say that it is still the same TV that it was before I made the repair. I think this is probably true in the case of a human being getting a transplant: if my heart were to wear out, I would cease to function as a human being, that is, I would die. However, if I get a heart transplant before my old heart ceases to function, I would presumably go on living and yet still be the same person as I was before the transplant.
I don't know how this would work in the case of the loaf of bread, because for one thing it is difficult to imagine what it would mean to say that there is a need to "replace" the flour that is in it with different flour. Maybe I made a wheat loaf when I was supposed to make an oat loaf or something, and I would like to take out the wheat flour and replace it with oat flour. But of course you can't do that, and even if you could it would not be the same kind of thing it was before, it would cease being a wheat loaf and it would become an oat loaf.
However that may be, the case of the Church seems clearly to be more like the TV than the loaf of bread, because the Church's parts, that is, the people that constitute her, can and do change, but the Church remains the same Church. Here is where a potential down-side to the notion of subsistence comes in. If we think of the Church as consisting of parts, we must think of the People of God as the constitutive parts of the Church. These people, taken as individuals, are very often sinners. This means that some of the Church's parts are sinful. The Church herself, however, according to dogma, is sinless. But this has not stopped some folks from asserting, rather wildly, that "the Church has sinned". What they mean, of course, is that some of the people in the Church have sinned, but because some of these people have been in positions of leadership in the Church, we now have a language of "institutional sin" that attributes sinfulness to the Church herself rather than her parts.
This is clearly a fallacy. In particular, it is the fallacy of combination: attributing to a whole one of the properties of some or all of its parts. For example, it may be that this particular nut or this particular bolt is very inexpensive. Here is a pile of such nuts and bolts, called a Lexus. The nuts and bolts are inexpensive, therefore the Lexus is inexpensive. Here is a sinner; there is a sinner; here is a pile of sinners called "The Church". These sinners are all sinful, therefore "The Church" is sinful. As it happens, the Church herself is not sinful, even though virtually all of her constituent parts are sinful. This is not a paradox: there are plenty of wholes that do not share some particular property with any of their parts. The set of integers is infinite even though each and every one of its members is finite. But even though this situation is not a paradox, some people find it impossible to imagine, and so they deny it. Because the Church is filled with sinful people, the Church herself must be regarded as sinful in some sense.
Now, it must be admitted that Ecclesia Dei semper reformanda est, but this does not mean that it is ever to be reformed because it is ever sinful. Sadly, the introduction of language of subsistence, bringing along with it the complicated mereology that is our ecclesial doctrine, has led to the mistaken notion that the Church is in itself sinful, which is clearly a heresy. The heresy can be avoided if the language of identity is used instead of the language of subsistence, so it remains a live question whether the language of subsistence is "better" in every respect than the language of identity, but I think that in most respects the language of subsistence is better, and for the reasons Mike gives. What one is to do with the boneheads who adopt either the heresy of a sinful Church or the heresy of a traditionalism gone badly awry is a difficult problem, but as long as the Church teaches anything there will be those who either don't understand the teaching or who willfully misrepresent it, so we are not entering frighteningly new territory here.