Answer: Prior to June 2008, many things. After June 2008, nothing.
Why June 2008?
In June 2008, the Statute of the Neocatechumenal Way (NCW) received final approval.
The Statute of the NCW is its charter, it's "birth certificate". Like any legally constituted association, that association only legitimately exists when it is in compliance with its charter. Outside the charter, it ceases to exist, either ipso facto or or by a formal sentence.
Prior to to the promulgation of its final charter, the NCW, like many associations within the church, operated at will and with the permission of this or that bishop or pastor. This is also how most religious orders start.
As the NCW grew, the growth necessitated greater organizational structure and eventually a charter from Rome.
In 2002, Pope John Paul II initiated a five year period of "ad experimentum" in which the practices and teachings of the NCW and its founders were scrutinized. The aim of the study was to bring full legitimacy to the particular "way" of living and sharing the Catholic faith.
To begin the process, the leadership of the NCW was instructed to formulate its own Statute and submit it to Rome for review. The leadership did this in 2002, and the 2002 Statute can be read here.
Rome spent five years studying the practices and teachings of the NCW and amending the 2002 Statute to bring it into conformity with what the magisterium of the Church required.
The product was the Statute of 2008 which you can read here.
In addition to promulgating the Final Statute, Rome spent an additional three years studying the teachings of its founders in a separate document called a "catechetical directory." Likewise, the directory was amended to bring it into conformance with the what the magisterium required to legitimate it, and the amended directory received final approval in 2011.
At this point, it is important to clarify the term "approval" because it is the word that the NCW leadership has taken to legitimate their every word and action, including those words and actions which depart from their charter.
The "approval" is ONLY for what is in the actual documents which have been approved, NOT for anything else. It is not a blanket approval of the NCW or its leadership.
And this is where the bishops come in.
The Catholic Church is an apostolic church. It's governing authority passes down through the ordained and consecrated successors of the apostles.
Thus when Rome magisterially promulgates any sort of instruction, directive, or, as in the case of the NCW, its constituting charter and its catechetical directory, the bishop of every diocese becomes the guardian of that instruction.
"Rome", meaning the central teaching authority of the Catholic Church whether it be the pope himself or a magisterial governing body at his service such as a congregation, dicastery, or pontifical council.
And in the case of the final approved version of the NCW Statute (2008), this fact is made perfectly clear at the outset of the document:
The Neocatechumenal Way is at the service of the bishop as one of the forms of diocesan implementation of Christian initiation and of ongoing education in faith. (Art. 1. §2)
As we have seen in recent scandal after crisis after crisis after scandal, most of the problems in the church can be traced to bishops who are unwilling to do their job. That's a nice way of saying "disobedient bishops".
And in Guam, we have one.
This has been demonstrated over and over whenever Archbishop Apuron violates liturgical norms and specific restrictions (as called for in Redemptionis Sacramentum, 74) by personally inviting lay members of the Neocatechumenal Way to come up during the time reserved for the homily to witness to their neocatechumenal experience and to promote it to a captive congregation.
This is an abuse of episcopal power, an abuse of the people, and a direct act of disobedience to the teaching Magisterium of the Church which has set liturgical norms that do not permit lay persons to take the pulpit during the time reserved for the homily.
I personally have sent more than one letter to Archbishop Apuron asking him to please provide evidence of his authority to do this. There has never been a reply.
Archbishop Apuron's blatant disobedience to Church authority (barring any evidence of a permit to do this) is also demonstrated time and again every time he celebrates the eucharist in his neocatechumenal community wherein he delays his own communion until all have received and permits the communicants to consume the sacred host seated.
It was observed at the 9:30am Mass this past Sunday at the Cathedral Basilica that Archbishop Apuron, after going to a wheelchair bound person to give her communion, proceeded to give communion to others who were standing in the front row still in their pew. Apparently he quickly realized that this was not a neocatechumenal eucharist and then returned to the sanctuary.
And as bishop, he is responsible for this sheer act of disobedience to the magisterial authority of the Catholic Church every time the same practice is repeated in any neocatechumenal celebration of the eucharist in his diocese.
There is no need to "catch" him doing this. On December 15, 2014, during a pastoral visit to St. Francis parish in Yona, he not only openly admitted that this is his practice, he claimed that Rome had approved it:
"Rome has approved it – even the way we’re receiving Communion. That they receive the host standing and they can sit down and everybody who receives it and they eat together...Rome has given permission for that. And it’s somewhere I need to find out where exactly, but they told us that permission is given."
Of course, Archbishop Apuron has never produced the permission which he said was "somewhere" and just needed to find it. However, the statement by itself should be a serious alarm to our apostolic visitors since the bishop of a diocese, the man deputed to be the guardian of our highest act of worship, does not know "where exactly" the approval for this very obvious departure from the sacred liturgy is.
In fact there is no approval and Archbishop Apuron knows this, In fact this is a bold faced lie to the parishioners of St. Francis Parish. In fact, this is the same lie he has been telling us since he said it on the radio in 2006 and caused a firestorm of protest.
But perhaps "lie" is a bit harsh. In fact, for him, it is probably no longer a lie. He was told this by his neocatechumenal superiors - the ones he actually obeys - and he has parroted this for nearly a decade. And therein lies the problem. This bishop simply obeys a different authority.
So, no. There is nothing "wrong" with the Neocatechumenal Way in so far as it is in compliance with its Final Statute and the approved version of its directory. However, there is everything wrong with a bishop who has no regard for either or for the authority of the Church from which his own authority is derived.
Final note: The faithful of the Archdiocese of Agana, in charity and out of regard for his office, have overlooked these abuses by their bishop for many years. It was not until Archbishop Apuron began publicly abusing priests who were unwilling to violate the same church norms that Archbishop Apuron was ordained and consecrated to uphold that the people began to protest against his violations of basic human dignity and blatant abuses of his office.