Between Protestants & Catholics what are the major disagreements with respect to Church history?

This is such a massive question, involving many issues, I doubt if anyone could even provide a summary list with dates and topics of difference. On this site, there just is not room to give a full answer. The reading of various books is really the best way to jump into this very deep pool. I would recommend the following:

The Pilgrim Church by E.H. Broadbent, a Penguin Classic first published in 1931, with 9 reprints. He traces various groups of believers in every generation (who avoided ritualism and sacerdotalism) from the 2nd century through till 1930, traveling widely over many years to collate often obscure information and statistics. This is a Protestant view.

The History of Christianity - a Lion Handbook (many illustrations and charts) 1977, written by a wide variety of accredited historians; non-sectarian

Encyclopedia of Theology, edited by Karl Rahner, Burns & Oates 1981, sections on Apostolic Succession / Fathers, Calvinism, Church History, Protestantism and the Reformation - all from a Catholic point of view.

The Pelican History of the Church - six paperback volumes by various authors of different denominations, dealing with the 2nd century through till 1964, with on-going reprints.

One brief answer to your question is that if Catholics feel entitled to trace their origins back to the first century, then the same leeway should be granted to Protestants. It ought to go without saying that the Bible must provide the only authentic mirror to hold conflicting claims up against, and that historical claims - in and of themselves - cannot prove which group is truly heir of the first century Church. The test can only be whether "they continued steadfastly in the apostles' doctrine and fellowship, and in breaking of bread, and in prayers" (Acts 2:42). Only those ones will have history on their side, showing that they have, indeed, "earnestly contended for the faith which was once delivered unto the saints" (Jude verse 3) - no additional doctrines, in other words.

This means that you cannot answer this question without comparing the two group's doctrines with the doctrines that had been delivered to the Church by the end of the first century, thus you must be deeply versed in biblical doctrine and practice. And the paramount doctrine to be utterly clear about is this one - what is the gospel of Jesus Christ? Get the biblical one and you have the biblical Christ. Get a man-made (i.e. corrupted) gospel, and whatever group is teaching that is never going to be the true heir to the apostles' teaching. That is the major point of disagreement between Protestants and Catholics, I would suggest.

OP: I’m not asking who’s right, but what are the major disagreements between the two camps.

The major differences between Catholic and Protestant such as the roles of the papacy vs elders, the differentiation between initial grace with subsequent meritorious works (so-called) vs the fullness and sufficiency of the grace of God and works sourcing to Him, the contrasts between the sacerdotal priesthood and priesthood of believers, and other conflicts all source to the first couple hundred years of Christian history. Specifically, it sources to how Christian Pentecost is understood by them.

Which group teaches that the believer has come to Zion and which teaches you have come to Sinai?

Sinai instilled the fear of God in them, raised up the sacerdotal priesthood, and more. For example, if one fears God and trembles at His voice, then one might want not just an intermediary such as a sacerdotal priest, but also someone in heaven who is nearer to God than thee.

(For they [at Sinai] could not endure that which was commanded, And if so much as a beast touch the mountain, it shall be stoned, or thrust through with a dart: Heb 12:20

Zion, however, is the complete opposite.

But ye are come unto mount Sion, and unto the city of the living God, the heavenly Jerusalem, and to an innumerable company of angels, Heb 12:22

Jesus is the sole mediator at Zion (Heb 12:24).

At Sinai, they offered the blood of others continuously. At Zion, Christ offered His own blood once.

Nor yet that he [Jesus] should offer himself often, as the high priest entereth into the holy place every year with blood of others; Heb 9:25

At Zion, each believer is a priest and king within the priesthood of Christ who is the King of kings.

At Sinai, it is the opposite.

So, to answer the OP, the major disagreements of roles, ranks, and results all source to a difference in view of what it means to come to Sinai or Zion. This difference formed very early, most likely sourcing to Josephus' change of heart at 70AD once he moved to Rome.

EDIT TO ADD.

The point of contrasting Sinai and Zion is to provide a basis for the history of Christianity from very early (circa 100-500) in the development of the Roman Catholic Church at Sinai to the Reformation that began to reestablish what it means that believers come to Zion.

The solas from works. The priesthood of believers from sacerdotal, exclusive priesthood. etc

Clarifying the question under Ecclesiology

The question can be improved by clarifying what you mean by "with respect to church history". This answer assumes you are asking Protestant vs. Catholic difference in understanding:

1. Authority of the visible church bishops and the Pope to teach doctrines in the past 2000 years; and
2. How Jesus is present in the church throughout history, following the aphorism of the first century writer Ignatius of Antioch: "Wherever Christ is, there is also the Catholic Church" (source: Epistle of Ignatius to the Smyrnaeans).

In systematic theology, this is covered under the topic of Ecclesiology where Protestants have their view and Catholics have their own view as well.

Both aspects of Ecclesiology are foundational because:

1. Doctrinal teachings flow from a certain understanding of how church history mediates tradition
2. Sacramental role of certain members of the body of Christ being separated from the rest (like how the Levites were chosen) flow from a certain understanding of how Jesus is visibly present in His body throughout history.

Foundational areas of agreement

1. True teachings (doctrines) are based on a combination of

   • apostolic interpretation of Scripture (as opposed to heretical interpretation of Scripture) AND
   • a certain preference of the theologies of the early church fathers & later theologians to work out further the teachings embedded in the canonized written scripture. Examples: the theology underlying the Nicene creed / the Westminster Confession, the theology of St. Augustine / St. Thomas Aquinas / John Calvin, etc.).

   This is with the understanding that those theologies are NOT revelation, but to clarify the right understanding of Scripture in areas that it doesn't teach explicitly: the doctrine of the Trinity, the exact nature of the humanity of Christ, Christ being the second Adam, which Old Testament commandments are still in force, clarification of the Sermon on the Mount, the role of Baptism and of the Lord's Supper, the meaning of the communion of saints, the continuation of certain spiritual gifts such as prophecy & healing, Christ vs. culture issues, Bible vs. science issues, theories of atonement, the experience of souls post death, etc.

2. The church was founded by Jesus as the head, spread by the 12 apostles empowered by the Holy Spirit starting at Pentecost, and continues until today as He works through the body to reconcile straying lambs into the fold through the hands and feet of the believers who bring the gospel to the end of the earth.

Foundational areas of difference

1. The role of apostolic succession as a visible expression of how the Holy Spirit works in preserving the apostolic teaching for believers in each period of history.

   • Protestants believe that the Holy Spirit solely works within the individual believer when reading the Scripture through individual study or when hearing the Scripture preached by competent pastors. It is possible because Protestants teach that Scripture possess the S.C.A.N. attributes (Sufficiency, Clarity, Authority, Necessity).

     Protestants also believe that only teachings explicitly taught in the canon are authoritative. Even the creeds have a lower authority than Scripture (cf. sola scriptura). Although Protestants believe that the Holy Spirit provides spiritual gifts to pastors, teachers, etc, their teachings (even if they are accepted over long periods by many people, such as those of Luther / Calvin) don't rise to additional authoritative teachings on par with Scripture.

   • Catholics believe that the Holy Spirit also works through the Magisterium to provide an authoritative voice to assist believers to interpret scripture faithfully according to what the original apostles taught. In addition, Catholics also see church history as the process of making explicit certain practices and teachings that were either part of the unwritten apostolic teaching or started in the first centuries of the church, such as purgatory, intercession of Mary and the saints, the growing recognition of the primacy of the Bishop of Rome, etc.

     Because Catholics believe that the Holy Spirit is behind those practices (working through the Magisterium), those teachings were not seen as non-Biblical accretions, since none of them contradict what the Bible teaches. In fact, the canon itself was a fruit of this process!

2. The sacramental role of the priesthood in addition to the priesthood of all believers.

   • Protestants don't believe that Christ appointed a vicar or a ministerial priesthood. In other words, there is no separate channel of grace other than the Holy Spirit directly to individual believers.

   • Catholics believe in visible administration of grace through specially called members of the body, who then became priests, bishops, and Popes. The 7 sacraments are a means of Jesus being visibly present to the faithful. The bishops and the Popes (through the apostolic succession through the laying of hands) hold the keys to the kingdom given by Jesus himself (Matt 16:17-19, Matt 18:15-19, John 20:19-23, etc).

3. Although both Protestants and Catholics saw the church started in the first century,

   • Protestants see church history as showing how the visible church was at times corrupted by her purported successors of the apostles to the extent that they became false teachers who obscured the true gospel, cf. Martin Luther's "rediscovery" of true gospel as Justification by Faith alone (sola fide). Thus they are unable to see the visible Roman Catholic church as coterminous with the true church. They then see the 16th century Reformation as a restoration of the true church, by creating a criteria (depending on the denomination, such as one by Calvin below) to be the boundary marker of the true spiritual church that is not coterminous with the many visible Protestant churches. Thus, although the visible Protestant churches started only in the 1500s, they all trace their spiritual lineage to the true spiritual church founded by Jesus. Protestants believe that a remnant of this true church continued under the corrupt visible Roman church up to the 1500s.

   • Catholics admitted that some of their bishops and Popes were sinful and corrupt in their personal lives but the Holy Spirit will never abandon the visible Roman Catholic church. The Holy Spirit:

     1. guarantees the efficacy of the sacraments up until today

     2. keeps giving the keys of the kingdom to the true successors of the apostles up until today

     3. keeps teaching through the Magisterium in addition to directly in a believer's heart

     Please note that the proper understanding of using the keys is that the Holy Spirit initiates and the key holder merely visibly exercises the authority. It's not as if the Holy Spirit can only administer forgiveness through the priests, nor that any arbitrary action exercised by a bishop / Pope carries the Holy Spirit's backing simply because they possess the keys.

Further study

1. Allister McGrath in his widely used textbook Historical Theology, 2nd Ed (2013) in chapter 4 (The Modern Day, 1750 to the Present Day) has a case study 4.5 Twentieth-century discussions of the doctrine of the church where he discusses 3 interpretations of Ignatius of Antioch's aphorism "Wherever Christ is, there is also the Catholic church":

   1. Christ is present sacramentally: This is the Catholic church position as being further clarified in the Second Vatican Council through council documents such as Lumen Gentium. Quote from Henri de Lubac:

      If Christ is the sacrament of God, the church is for us the sacrament of Christ; she represents him, in the full and ancient sense of the term, she really makes him present. She not only carries on his work, but she is his very continuation, in a sense far more real than in which it can be said that any human institution is its founder’s continuation.

   2. Christ is present through the Word: This is the 16th century reformers' position, especially Calvin in his classic statement on the nature of the church:

      Wherever we see the Word of God purely preached and listened to, and the sacraments administered according to Christ’s institution, it is in no way to be doubted that a church of God exists. For his promise cannot fail: “Wherever two or three are gathered in my name, there I am in the midst of them” (Matthew 18: 20). … If the ministry has the Word and honors it, if it has the administration of the sacraments, it deserves without doubt to be held and considered a church.

   3. Christ is present through the Spirit: This is the view of the Protestant Karl Barth, liberation theologians and Orthodox theologians. A quote from liberation theologian Leonardo Boff:

      The church comes into being as church when people become aware of the call to salvation in Jesus Christ, come together in community, profess the same faith, celebrate the same eschatological liberation, and seek to live the discipleship of Jesus Christ. We can speak of church in the proper sense only when there is question of this ecclesial consciousness.

2. A formal study comparing Protestant vs. Catholic view of church history can be read in Chapter 6 (Protestant and Catholic Histories of the Early Church) of the 2003 book Historical Method and Confessional Identity in the Era of the Reformation (1378-1615) edited by Irena Backus, a professor of the History of the Reformation. Portions of that chapter can be read from Google Book preview which discusses contrasting ways that Protestant vs. Catholic church historians support their respective ecclesiology by their method of selecting certain actions of the early church authorities and certain teachings of the early church fathers as proof.