What are the main doctrines of the New Apostolic Church, and what are the big differences in their teachings from the perspective of Evangelical Protestants?
The difficulty in arriving at a meaningful comparison between the NAC and Protestant Christianity (whether they are evangelical or not) is that the NAC declare they are neither Protestant nor Catholic, even though they came out of the Catholic Apostolic Church.
The Catholic Apostolic Church was a religious movement which originated in England around 1831 and later spread to Germany and the United States. While often referred to as Irvingism or the Irvingian movement, it was neither founded nor anticipated by Edward Irving. The church was organised in 1835 under the lead of self-proclaimed apostles. Source: https://en.wikipedia.org/wiki/Catholic_Apostolic_Church
Readiness for an expected immediate Second Coming of Christ was the central aim of the congregations; the restoration of perfect institutions by the Apostles was deemed necessary to prepare the whole church for this event. The doctrines of achievable personal holiness, attainable universal salvation, the true spiritual unity of all baptized persons, living and dead, in the 'Body of Christ', the possibility of rapture without dying, and the necessity of the fourfold ministry directed by Apostles for perfecting the Church as a whole, formed the cornerstones of the theology. Source: https://en.wikipedia.org/wiki/Catholic_Apostolic_Church#Theology_of_the_Second_Advent
Like the Catholic Apostolic Church, the New Apostolic Church believes in apostolic succession, that their leaders are the successors of the twelve apostles. Most evangelical Protestant churches refute this teaching although I have been informed that "Anglicans, Presbyterians andsome Lutherans and Pentecostals trace theirordinations back to the early church and apostles." I don't know if that means they believe in apostolic succession, however, in the same way the Catholic Church does.
It is the view of most evangelical Protestants that apostolic authority was passed on through the writings of the apostles, not through apostolic succession. See Acts 17:10-12; 20:32; 2 Timothy 3:16-17). This is the ‘Scripture alone’ sola of the Protestant Reformation. See: https://www.gotquestions.org/sola-scriptura.html
The New Apostolic Church believes they have restored the continuity of the early church. Protestant churches refute this claim. Here is an extract from an evangelical Protestant source:
[Restorationists] have in common that true Christianity had died out many years ago and it needed to be restored to its original New Testament form. Some of these groups believe they alone are the embodiment of true Christianity, some going so far as to teach that all other groups, including mainline Protestant denominations, are not really Christians at all, having lost their way over the centuries to complete apostasy. They are convinced that the drift from Christian principles is so extreme as to render the church irredeemable, and, therefore, it must be completely rebuilt. https://www.gotquestions.org/restorationism.html
A distinctive and controversial feature of the New Apostolic Church is its doctrine about the dead. A special divine service of remembrance and intercession in prayer takes place three times annually. In such divine services the apostles also administer the sacraments to a living subsidiary minister. Regarding this, New Apostolic Christians refer to several biblical texts such as 1 Corinthians 15:29. There are no limitations to whom in the beyond may be baptised by apostles. Source: https://en.wikipedia.org/wiki/New_Apostolic_Church#Beliefs_about_death_and_beyond
Protestant churches do not consider prayers or baptisms for the dead to be biblical. Here is an extract from an evangelical Protestant source on praying for or baptising the dead:
Praying for the dead is not a biblical concept. Our prayers have no bearing on someone once he or she has died. The reality is that, at the point of death, one’s eternal destiny is confirmed. Either he is saved through faith in Christ and is in heaven where he is experiencing rest and joy in God’s presence, or he is in torment in hell. The story of the rich man and Lazarus the beggar provides us with a vivid illustration of this truth. Jesus plainly used this story to teach that after death the unrighteous are eternally separated from God, that they remember their rejection of the gospel, that they are in torment, and that their condition cannot be remedied (Luke 16:19-31)... The writer to the Hebrews tells us, “Just as man is destined to die once, and after that to face judgment” (Hebrews 9:27). Here we understand that no change in one’s spiritual condition can be made following his death—either by himself or through the efforts of others. https://www.gotquestions.org/praying-for-the-dead.html
If the NAC practice baptism for the dead (like the Church of Jesus Christ of Latter Day Saints) then this would be denounced by Protestant churches. I do not know if this is what happens in the NAC, so I will simply leave a link to an article that briefly discusses this: https://www.gotquestions.org/baptism-dead.html
Regarding conversion, however, the NAC belief that water of baptism is an essential part of rebirth and entitles the believer to the sealing of the Holy Spirit is not the view of most evangelical Protestant churches. The NAC view that the Holy Spirit is given by the act and authority of an apostle, which makes the believer a child of God and incorporates him into the body of Christ is not the view of most evangelical Protestant churches.
Conclusion: The doctrines of the New Apostolic Church bear similarities to many evangelical Protestant churches. The virgin birth, sinless life, and atoning death of Jesus Christ, the need of personal repentance and confession for forgiveness of sins, and the literal return of Jesus Christ to earth are all held by this church.