I read about Anglo-Catholicism and what I found is that there aren't differences between them and Catholic (in term of doctrine). Am I wrong?
If I'm not in wrong, the only difference is in the Pope's authority vs the authority of the Archbishop of Canterbury (or the Queen of England), so why they do not come together?
I can't understand the schism.
Even with some changes within some of the Anglican community (the Confession of St Louis, establishment of the Traditional Anglican Communion, or the Anglican Ordinariate per the link in @Geremia's answer) the differences are rooted in the original schism.
The schism itself had as much to do with politics and culture as it did religion. That topic is worthy of a separate question, but I'll summarize it here.
The divorce/annulment requested by King Henry VIII wasn't the first time a monarch had applied for such a dissolution of a marriage. The rejection (in a large part due to the queen, Catherine, of the Spanish royal family, being aunt of Charles V, a Hapsburg) acted as a catalyst to tensions already present in European/Christian culture. The Reformation was already underway, and Humanism was on the rise with the Enlightenment coming soon.
It is somewhat ironic, in retrospect, that King Henry whose publication in 1521 of Assertio Septem Sacramentorum ("Defence of the Seven Sacraments") earned him the title of Fidei Defensor (Defender of the Faith) from Pope Leo X led the break from Rome.
Between the mid-1530's -- despite protests in England against this break -- and 1542, when England's remaining monasteries were all dissolved and their property transferred to the Crown, the breach became irreconcilable. None of Henry's successors found it in their interest to heal the wound1. A shame, but that's politics for you.
Not much, but it gets sticky in particulars.
The core significant differences between the Anglican and Catholic confessions -- seen from a Roman Catholic point of view -- is that once the break was made by the clergy then subsequent ordination of clergy was/is invalid in the eyes of the Holy See in Rome. (This is related to the Archbishop of Canterbury being excommunicated and reconciliation not happening). This disputation of ordination is based on the doctrine of Apostolic Succession, which holds that ordination is a sacrament passed down from Peter to the present pontiff.
CCC 815 What are these bonds of unity? Above all, charity "binds everything together in perfect harmony." But the unity of the pilgrim Church is also assured by visible bonds of communion: - profession of one faith received from the Apostles; -common celebration of divine worship, especially of the sacraments; - apostolic succession through the sacrament of Holy Orders, maintaining the fraternal concord of God's family.
So how did the sacrament of ordination become an issue? The detailed reason (and the ensuing loss of apostolic succession) went beyond the schism itself — Eastern Orthodox ordinations are held as valid by the Catholic Church. It is derived from the fact that the Anglican rite of episcopal ordination (ordaining bishops) was altered to remove any mention of the Mass as a sacrifice. This was deemed by Pope Leo XIII to be a change that vitiated the form of the sacrament of Holy Orders. (Preceding paragraph owes a deep debt to @AthanasiusOfAlex).
If the ordination is invalid -- again, as seen from the Roman Catholic perspective -- then much else gets called into question. At the least the two churches are no longer in communion. That means that receiving communion as a sacrament isn't recognized across boundaries, and "the real presence" via transubstantiation won't happen when celebrating the Mass since the priest (Anglican) isn't acting in Persona Christi Capitis. (This an example of what Lumen Gentium refers to as being in imperfect communion vis a vis other Christian Denominations). As @brasshat points out, the Anglican confession does not accept transubstantiation as a doctrine, nor as a truth.
In the person of Christ the Head . . .
CCC 1548 In the ecclesial service of the ordained minister, it is Christ himself who is present to his Church as Head of his Body, Shepherd of his flock, high priest of the redemptive sacrifice, Teacher of Truth. This is what the Church means by saying that the priest, by virtue of the sacrament of Holy Orders, acts in persona Christi Capitis:
This point on ordination is a non-trivial obstacle.
CCC 1538 Integration into one of these bodies in the Church was accomplished by a rite called ordinatio, a religious and liturgical act which was a consecration, a blessing or a sacrament. Today the word "ordination" is reserved for the sacramental act which integrates a man into the order of bishops, presbyters, or deacons, and goes beyond a simple election, designation, delegation, or institution by the community, for it confers a gift of the Holy Spirit that permits the exercise of a "sacred power" (sacra potestas) which can come only from Christ himself through his Church. Ordination is also called consecratio, for it is a setting apart and an investiture by Christ himself for his Church. the laying on of hands by the bishop, with the consecratory prayer, constitutes the visible sign of this ordination
It is worth noting that in this passage by "Church" and "his Church" the meaning is "The Roman Catholic Church" and all those in communion with them (like the Easter Rite Catholics ...).
On the other hand, Anglican Baptism is a valid sacrament, as it retains water and the Trinitarian form.
On the bright side, Popes Benedict XVI and John Paul II put some energy into working toward a resolution of the differences between the two confessions. Much goodwill has been offered as well among the Anglicans.
Is there hope of reconciliation? Yes.
When? Unknown.
1 Well, the Church of England was briefly reunited with the Roman Catholic Church under Queen Mary, before separating again under Elizabeth I, Henry's daughter by Anne Boleyn.
Granmirupa, in fact you are wrong. There remain considerable differences between most who consider themselves "Anglo-catholic" and those who self identify as Catholic, other than whether the Archbishop of Canterbury, or the Bishop of Rome exercises authority over the church. One of the differences between [Roman] Catholics and Anglo-catholics is that it's much easier to define who is a Roman Catholic. Simply put, a Roman Catholic is a person who accepts the authority of the Bishop of Rome, and accepts the correctness of those doctrines and dogmas, like the doctrine of transubstantiation, the Assumption of the Mother of God, the doctrine of purgatory, and the church's rules regarding the celibacy of the Clergy, and the existence of seven sacraments, that the Catholic church demands that the faithful believe.
An Anglo-Catholic, especially in the present times, is a much more nebulous identification. The present use of the term dates only as far back as the last 3/4 or 1/2 of the nineteenth century, when certain Divines in the Church of England began to reconsider, and recover some of the customs and ceremonial of the Roman Catholic Church, and began to develop greater appreciation for some parts of Roman Catholic theology. Some of these individuals, including John Cardinal Newman and Henry Manning, joined the Catholic Church, but others, like John Keble and Edward Pusey continued in the Anglican Communion, and continued to subscribe to the thirty nine articles, which rejected the doctrines of transubstantiation, and purgatory, denied the supremacy of the pope, the necessity of unmarried clergy, held that there were two, and not seven, sacraments, and denied that the Apocrypha (deutero-canonicals) were to be included in the Canon of Holy Writ. Also, at the time of the origin of the Anglo-catholic movement, the Roman church still insisted that all liturgy be in Latin, while the 39 articles required that Anglican liturgies be in a language that the people could uncerstand. The result was a series of volumes such as the Anglican Missal, which incorporated most of the form and structure of the Mass, but in English, and with some additions to the prayer of consecration in the English prayer book. During this period, Briggs and Frere produced a psalter which helped re-introduce plainsong tunes (using Sarum, rather than Latin rite endings) for singing the Psalter, and other canticles. And while Anglo-catholics recovered some of the practices relating to rites that the Roman Church considered sacraments, like extreme unction, confessions, and holy orders, the understanding of these rites was never changed so that they were considered sacraments; article 25 (of the thirty nine), still explicitly is the official position of Anglicans, that
There are two sacraments ordained of Christ our Lord in the Gospel, that is to say, Baptism, and the Supper of the Lord
Those five commonly called sacraments, that is to say, Confirmation, Penance, Orders, Matrimony, and extreme Unction, are not to be counted for sacraments of the Gospel...
Anglo-catholiism was never universal in any branch of Anglicanism. In most dioceses, there might be a number of Anglo-Catholic Parishes (sometimes only one or two), while most of the rest of the parishes and missions in the dioceses were Anglican (or in the US, Episcopalian), at the same time as in other dioceses, there might be a larger proportion of Anglo-catholics, and a smaller number of those who were not.
More recent events have cast made the situation even more confusing. There are a small number of Anglo-catholic parishes which are socially progressive, so that they support the Ordination of Women to the clergy, and perhaps other recent social movements. A number of others who self-identify as "Antlo-Catholic" have rejected some of these recent developments, and have joined one of the "continuing" groups. Further, some in the Anglo-catholic movement have come to hold views more aligned to the Roman Catholic church in areas such as the doctrine of purgatory, and some of the more recent Marian developments (e.g., the dogma of the Immaculate conception of the Mother of God), though this is not the official position of any Anglo-catholic group of which I am aware [Note: I am not meaning to represent that there are no Anglo-catholics who hold these view; I'm just not aware of them.]
As to the recent developments of the Anglican personal ordiariate, note that this affects only customs and ceremonies of the liturgy. Presumably, those choosing to take advantage of the terms of Benedict's ordinariate, are still required to adopt the doctrines to which the Catholic Church demands all adherents of the western rite subscribe, such as transubstantiation, and the supremacy of the Bishop of Rome.
So the reasons Anglo-catholics (or at least the Anglo-catholic authoring this answer) do not come together with those who profess the supremacy of the Bishop of Rome, is that there are several areas where there is great difference of opinion about matters of the Faith and the Church. I would also note that since the changes in Roman Catholic theology that have occurred since World War II, (used, not because I mean to suggest that war had much to do with the changes in theology itself, but mainly because it provides a convenient marker for the period of time), and especially since the second Vatican council, do seem to make the differences between Anglo-catholics on the one hand, and Catholics, on the other, at least seem less than they once did.
