I do not intend to explain what this is all about ab initio to those who do not already know the general outlines. Just to add some facts which those who do know may not be familiar with.
In 1995 the Pontifical Council for the Promotion of Christian Unity published a learned and interesting paper which suggested that a lack of correlation between the Greek (ekporeuesthai) and Latin (procedere) terms for "proceedeth" is part of the problem. ekporeuesthai refers to the origin of the Holy Spirit within the eternal and glorious economy of the Holy Trinity. And, since the Father is the Source (pege, aitia) of the being of the other two Persons, clearly the Spirit ekporeuetai from the Father alone. To suggest that he might ekporeuesthai from the Son as well is to posit two sources of Divinity and thus, in effect, to believe in two Gods.
Procedere, on the other hand, is a broader term. As well as sharing the meaning of ekporeuesthai, it also encompasses the Sending, wthin time, of the Spirit by the Son. When the Western Church was battling against Arianism, it seemed important to safeguard the full divinity of the Son by incorporating into the Creed His authentic Missio of the Spirit.
So you could argue that Filioque with ekporeuesthai is gravely erroneous because it is tantamount to polytheism, while procedere without the Filioque is dangerously suggestive of Arianism.
It is well known that Rome firmly forbids the addition of Filioque to the Creed when it is said in (or translated from) Greek - whether by 'uniate' Byzantines or in ecumenical contexts. But she has been slow to delete Filioque from the Creed when it is used in (or translated from) Latin.
However, in 2000 a very significant new development occurred. Cardinal Joseph Ratzinger issued a document called Dominus Iesus, which was immediately made the object of hysterical abuse by illiterates who couldn't read it (including poor dopey George Carey) - you probably remember the hooha raised at that time by the ecumaniac industry both inside and outside the Roman Catholic Church. This was unfortunate; the document represented some very interesting advances ecumenically in several respects [I pointed this out at the time and received a very gracious letter of appreciation, full of ecumenical hopes, from a member of the CDF, the then Cardinal Archbishop of Dublin]. Not the least of these is that it began by giving, as 'the fundamental contents of the profession of the Christian faith', the 'Nicene' Creed in Latin and without the Filioque.
This cannot be without significance. I am not one of Papa Ratzinger's regular boozing buddies, but I have a hunch about how he might think of operating. Take the SSPX 'schism'. He made two moves, the motu proprio and the rescinding of the excommunications, in response to positive gestures from the Lefebvreist side. I wonder if he is saving up the elimination of the Filioque from the Latin Rites to be part of a significant combined two-steps-forward by the Catholic and Orthodox sides together in this dialogue.