This decree is loaded with information, and so I will have to highlight a few of the elements of the Latin theology of procession in order to explain later why the Eastern Orthodox Churches cannot accept this definition as authoritative and binding. First, the decree proclaims the idea that the Father and the Son together give the Holy Spirit His subsistent being. Now in doing this, as the decree itself emphasizes, the Father and the Son act, not as two principles, but as a single principle, and this is important because there cannot be two principles within the Godhead, at least not without admitting that there are two gods. In other words, there can only be one source or font of divinity (πηγή τῆς θεότητος), and in the case of the procession of the Holy Spirit, the Latin Church holds that the Father and the Son together are this single principle or source (that is, for the spiration of the Spirit). Moreover the decree itself draws an equivalency between the terms cause and principle, asserting that the Latin understanding of the term principle is roughly the same as the term cause (αιτία) in Greek Trinitarian theology, and of course the Greek Church uses the word cause in connection with the Father alone, because it signifies His monarchy. 
Now it is hard to see how Byzantine – particularly Palamite – Triadology can be reconciled with the above formulation of the procession of the Holy Spirit. In Eastern Triadology the focus is always placed upon the monarchy of the Father, who, as the sole source of divinity, causes the hypostasis of the Son through generation, and the hypostasis of the Holy Spirit through procession (ἐκπόρευσις). Now, in fairness it must be admitted that the Latin tradition also holds that the monarchy of the Father is important, and it tries to safeguard this theological truth by saying that the Father is the "principle without principle" in the Trinity,  but this formulation opens up the possibility, and can even be said to imply the necessity, of a "principle with principle" in the Godhead, i.e., the Son. This idea of a secondary cause in the Trinity is foreign to the Eastern theological tradition, which has always focused upon the idea that there is, and only can be, one cause in the Trinity, the hypostasis of the Father. In other words, the idea that there could be two causes within the Godhead, an uncaused cause, and a caused cause, is utterly foreign to Byzantine Triadology; and moreover, within the light of Byzantine tradition the assertion of two causes in the Trinity smacks of polytheism, because there would be as many gods as there are causes of divinity.
The Latin Church's solution to the problem of two principles or causes within the Godhead is to assert that the Holy Spirit proceeds eternally from the Father and the Son – in the words of the Florentine decree – ". . . as from one principle and a single spiration."  But this solution causes additional problems, because as St. Gregory Palamas taught, generation and procession are hypostatic properties of the Father alone; and so, to posit the idea that the Son somehow shares in the existential procession of origin of the Holy Spirit confounds the hypostases of the Father and the Son, collapsing them into one and the same hypostasis.  The Latin Church, by asserting the idea that the Father and the Son form a single principle in the spiration of the Holy Spirit, has fallen into a form of Sabellian modalism, because both begetting and spiration are personal properties of the Father alone, and as personal (hypostatic) properties, they cannot be shared with any other person in the Trinity, or the real distinction between the hypostases collapses.
One further difficulty results from the Latin doctrine which holds that the Father and the Son form a single principle in the spiration of the Spirit, and it is focused upon the nature of the unity of the Godhead. It is an ancient principle of Catholic Triadology that anything that is common to two of the hypostases of the Trinity, is common to all three hypostases, because of their common essence (οὐσία); in other words, if the Father and the Son are a "single principle" in the spiration of the hypostasis of the Holy Spirit, it follows that the Spirit must also be a "single principle" with them in His own spiration, and that is clearly nonsensical. The hypostases of the Trinity are only distinguished by their unique hypostatic properties (idiotes), and so anything that is common to the Father and the Son, must also be common to the Holy Spirit. As St. Basil said, "The Spirit shares titles held in common by the Father and the Son; He receives these titles due to His natural and intimate relationship with them." 
Thus, the idea that the Father and the Son can be a "single principle" in the spiration of the hypostasis of the Holy Spirit involves a confusion of hypostasis and essence (οὐσία) within the Godhead, because anything common to the hypostases is founded upon the one divine essence (οὐσία) that they share, and that is why the Western notion that the Father and the Son can be a "single principle" in the procession of the Holy Spirit's hypostasis is theologically unworkable. Therefore, to hold that the Father and the Son can be a "single principle" of origin in relation to the hypostasis of the Holy Spirit involves either Sabellian modalism, or an essential subordination of the Spirit to the Father and the Son, because He does not possess a common quality shared by the Father and the Son, and consequently must be essentially distinct and subordinate in relation to them.
Sadly, these differences are not the only source of difficulties separating Eastern and Western Trinitarian thought, because with the rise of Scholastic philosophy in the West a new Aristotelian theory of the Trinity was devised, which involved the reduction of the hypostases to mere relations of opposition within the divine essence, yet there can be no “opposition” within the Godhead, since God is adiastemic. Added to the problems associated with this new philosophical theory of the Trinity was the fact that it was often couched in language related to the Augustinian psychological metaphor for explaining the distinctions between the persons within the Godhead according to processions of intellect (i.e., for the Son) and will (i.e., for the Spirit), even though the divine intellect and will are common properties of the triad of divine hypostases. Now, of course, the Eastern Fathers never accepted this line of philosophical reasoning, and instead affirmed that the hypostases of the Trinity are different in their manner of subsistence (τρόπος υπάρξεως). Thus, the Son, as only begotten, receives His subsistence from the Father through generation; while the Spirit receives His subsistence from the Father alone by procession (ἐκπόρευσις), and generation and procession are ineffably distinct subsistent realities. That being said, there is no danger of division within God, because St. John Damascene's doctrine of perichoresis allows the distinct hypostases to indwell each other, while remaining truly distinct, and that is why the Spirit, which is properly the Spirit of the Father, is also the Spirit of the Son, but as St. John goes on to say, ". . . we do not speak of the Spirit as from the Son."  Clearly, there is no filioque in the theology of St. John Damascene, nor is there one in the theology of St. Gregory Palamas, and in fact both men directly reject the filioque as can be seen in the case of St. John from the quotation just given.
Moreover, St. John Damascene does not reduce the hypostases to mere relations of opposition within the divine essence as do most Western theologians (for example St. Thomas Aquinas), nor does he fail to distinguish between essence (οὐσία) and hypostasis as Westerners since the time of St. Augustine have tended to do.  Now as far as the Spirit's existential origin is concerned, both St. John and St. Gregory hold that it comes from the Father alone, proof of this can be found by looking at what Fr. Andrew Louth wrote in his book on Damascene, because as he indicates, St. John ". . . speaks of the Holy Spirit as 'the Holy Spirit of God the Father, as proceeding from Him, who is also said to be of the Son, as through Him manifest and bestowed on the creation, but not as taking His existence from Him.' (St. John, Sabbat. 4:21-23)."  St. Gregory Palamas also teaches this, for as he put it, the ". . . pre-eternal rejoicing of the Father and the Son is the Holy Spirit who, as I said, is common to both, which explains why He is sent from both to those who are worthy. Yet the Spirit has His existence from the Father alone, and hence He proceeds as regards His existence only from the Father."  Thus, the Father alone gives existence to the hypostasis of the Spirit, and there can be no existential filioque. 
The East's negative view of the filioque, as it is formulated in the West, does not mean that the East rejects the idea that the Spirit is manifested through the Son. In fact, at the Blachernae Council (A.D. 1285) the Byzantine Church taught that the Holy Spirit is manifested as (or in the) divine energy through the Son, but that this manifestation (φανέρωσις) of the Spirit does not involve the existential hypostatic origin of the Spirit, since that comes only from the Father.  This distinction (that is, between hypostatic origin and manifestation) is made by St. John Damascene, and other Eastern Fathers, but was clarified by St. Gregory of Cyprus at the Council of Blachernae.  In the dogmatic tome of the Council eleven anathemas were pronounced, but in my paper I am only going to briefly examine one of those definitions:
[Those] who affirm that the Paraclete, which is from the Father, has its existence through the Son and from the Son, and who again propose as proof the phrase "the Spirit exists through the Son and from the Son." In certain texts [of the Fathers], the phrase denotes the Spirit's shining forth and manifestation. Indeed, the very Paraclete shines forth and is manifest eternally through the Son, in the same way that light shines forth and is manifest through the intermediary of the sun's rays; it further denotes the bestowing, giving, and sending of the Spirit to us. It does not, however, mean that it subsists through the Son and from the Son, and that it receives its being through Him and from Him. For this would mean that the Spirit has the Son as cause and source (exactly as it has the Father), not to say that it has its cause and source more so from the Son than from the Father; for it is said that that from which existence is derived likewise is believed to enrich the source and to be the cause of being. To those who believe and say such things, we pronounce the above resolution and judgment, we cut them off from the membership of the Orthodox, and we banish them from the flock of the Church of God. 
First, the document says that: "The Father alone is the principle without principle of the two other persons of the Trinity."  Now, the problem with this statement is that the Father, rather than being described simply as the "principle of the two other persons of the Trinity," is described as the "principle without principle," which can imply that the Son is a "principle with principle" within the Trinity (i.e., that the Son is a secondary principle within the Godhead). The idea that there can be a "secondary" principle in the Godhead is contrary to the teaching of the Eastern Church, and would ultimately destroy the monarchy of the Father, replacing it with a diarchy of the Father and the Son.
Second, the "Clarification" states that: "The Holy Spirit, therefore, takes his origin from the Father alone (ek monou tou Patros) in a principal, proper, and immediate manner."  The problem with this statement is centered upon the concluding portion of the formula, that is, the part of the text that says that the Spirit comes from the Father alone in a "principal, proper, and immediate manner," because this modifying phrase implies, or at least allows for the possibility, that the Son is involved in the existential origin of the Spirit in a secondary, received, and mediate manner. This kind of secondary or mediate causation is incompatible with the Triadology of the Eastern Fathers, and in particular with the doctrine of the Cappadocians, because as St. Gregory Nazianzus said, ". . . all that the Father has belongs likewise to the Son, except Causality."  Now, in order for the ecumenical dialogue between the Roman Catholic Church and the Eastern Orthodox Churches to advance, the Latin Church is going to have to issue a document that cannot be read in an equivocal manner on these issues. In other words, it must say that the Father is the principle of divinity, period, end of sentence, and with no modifying phrases or clauses added on. Thus, the West will need to say that the Spirit proceeds from the Father, without adding modifiers like "principally, immediately, properly, etc.," which can imply that the Son Himself participates in the hypostatic origination of the Spirit.
We do not say that the Son is from the Father in as much as He is begotten by the divine essence, but rather in as much as He is begotten by the Father as person. For the essence is common to the three persons, but begetting is proper to the Father personally. That is why the Son is not begotten by the Spirit. Consequently the Spirit is also from the Father; He possesses the divine essence, proceeding from the person of the Father. For the essence is always and absolutely common to the three persons. Therefore the act of spiration is proper to the Father as a person and the Spirit does not proceed from the Son, for the Son does not have the personal properties of the Father. 
Sadly, the insertion of the filioque into the Niceno-Constantinopolitan creed shows that the West has confused two distinct – but inseparable – divine realities: (1) the existential procession of the Holy Spirit as person (υποστασις), which is from the Father alone; and (2) the Spirit's eternal manifestation as divine energy (i.e., as uncreated grace), which is from the Father through the Son. In other words, in the theology of the Eastern Fathers the Holy Spirit proceeds as hypostasis from the Father alone, but He is manifested – both temporally and eternally – from the Father through the Son, not as hypostasis, but as divine energy; and this energetic manifestation expresses the consubstantial communion of the three divine hypostases within the Godhead. Now, as is clear from what has been said, it is vital that the Spirit's energetic manifestation through the Son not be confused with His hypostatic procession of origin from the Father alone, because that would ultimately lead to Sabellian modalism.
It should be noted, of course, that these are only a few of the problems with the Vatican's "Clarification of the Filioque," and so, although it is a valiant attempt by the Western Church to make the filioque more acceptable to the East, it ultimately highlights the differences between the two sides as it concerns the doctrine of the hypostatic procession of origin of the Holy Spirit. Nevertheless, I do not want to give the impression that the document is an utter failure, because it at least shows that the West realizes that the filioque is a true obstacle to the restoration of communion, and that further dialogue on this issue will have to be carried out if there is to be any chance at all of resolving this doctrinal disagreement.
Clearly, the best solution put forward so far to resolve the problem of the filioque can be found in the Agreed Statement of the North American Orthodox / Catholic Theological Consultation, which put forward the recommendation that the Latin Church remove the filioque from all liturgical and catechetical documents.  The use of the original creed by the Latin Church in its liturgical celebrations, and catechetical instructions, would facilitate ecumenical dialogue, while simultaneously removing one of the major obstacles to the restoration of communion between the Roman Catholic Church and the Eastern Orthodox Churches.