The reactions of Christians to differences in doctrine were both theological and political. I'll deal primarily with the political since this is a history sub and only mention the theological in broad strokes at the end. Please let me know if this does not answer your question and I'll try to do so, but I am trying to step carefully and avoid theology as much as possible. If you prefer a more theological answer you may want to try r/AskBibleScholars.
After the period of major persecution under Diocletian, when Christians could finally be vocal and open as they argued doctrinal points, more controversies began to arise. The debate over Arianism started in 314 CE when Alexander, Bishop of Alexandria excommunicated Arius, a pastor for his Christological teachings on the nature of Christ. Arius found Bishops outside of Egypt who supported him and controversy spread throughout the Eastern Roman Empire via different councils and counter councils until Licinius, the ruler of the Eastern Empire, forbade them.
In 324, Constantine won his war against Licinius and united the Eastern and Western halves of the Roman Empire. Part of Constantine's validation for the war was Licinius' recent persecution of Christians in the Eastern part of the Empire. In 312 CE, Constantine had realized how helpful the church could be and had begun, appointing Christians to various roles. By 325, he realized that unity in the church would be an important boon to unifying his new Empire. Thus began the state role in religion that was particularly present in the Arian controversy. To that end, he sent letters to Alexandar the Bishop of Alexandria and Arius he told them that they' 'were quarreling over small, indeed exceptionally trivial points,' that they should behave in a civilized way like philosophers, who disagree on small matters while remaining in total agreement on basic principles; and that it was wrong to set brother against brother for the sake of 'a few pointless verbal disputes'
Constantine sent his religious advisor Ossius, Bishop of Spain, to investigate and probably to try to come to some consensus, but it did not work. Instead, Constantine put his political weight behind the call for a council that met in Nicea in 325. During the theological debate, Eusebius of Caesarea (originally in the Arian camp) set forth a formula that used mostly biblical language from the gospel of John. It did not use words for Jesus like "creature of God" or "product of God" as Arius did. However, it was, in the Alexandrian view, still left open to Arius's unorthodox interpretation. It is at this point that Constantine stepped in, saying that the formula sounded good to him and that it was the one that should be agreed upon with the addition of the word "homoousios" meaning "of one essence." This was the term that the Arians disagreed with the most; however, they were in the minority, and in the end, only three bishops did not sign it. Those Bishops, along with Arius, were sent into exile immediately.
However, the Council of Nicea did not really solve anything. Arius kept writing and teaching his views, as did Eusebius of Nicomedia, who was exiled shortly after Arius. But the men were eventually welcomed back once they proved their sufficient orthodoxy to Constantine. This may not have been hard, as some scholars suggest that Constantine wasn't that orthodox himself in this matter and was probably Arian or Semi Arian.
After Constantine's death, many of his successors were pro Arian, and it was Athunasisus (who became the bishop of Alexandria after Alexander's death and head of the anti-Arian camp) who was excommunicated five times.
As Grant notes, "The ups and downs of Eusebius of Nicomedia offer a prime example [of the politics at play]. First, he stood close to Licinius; then, Constantine denounced and banished him. Reinstated as bishop, he even served as tutor to the future emperor, Julian, perhaps teaching him something about survival. Constantine on his deathbed craved spiritual security, and it was Eusebius of Nicomedia who baptized him." The same thing happened to Athunasusus after Constantine's death. Politics were played on both sides.
It wasn't until 381, and the Council of Constantinople, that the Anti-Arians and Semi Arians were able to unite and the matter was put mostly to rest. The question as to why this took 56 years can probably be answered in large part by political maneuverings mentioned above. As Barnes notes, "Recent investigations have established that from the very start the so-called 'Arian controversy' has a close structural resemblance to modern party politics. Two cohesive groups of eastern bishops and theologians were pitted against each other and, although not all members of either group by any means shared identical theological views or perspectives, they nevertheless cooperated with other members of their group in acting together in ecclesiastical politics in opposition to the other group."
Obviously, all the parties felt strongly that understanding Jesus's relationship with God was important. From a theological standpoint, what a Christian believes about who Jesus is, and his relationship to God affects the understanding of many important doctrines within the Christian faith. This is why, to the churchmen, a compromise was not as easy as Constantine made it sound in his letters.
Sources:
Barnes, Timothy David. 2014. Constantine: Dynasty, Religion, and Power in the Later Roman Empire. Blackwell Ancient Lives. Chichester, West Sussex, UK: Wiley Blackwell.
Grant, Robert M. "Religion and Politics at the Council at Nicaea" The Journal of Religion, Vol. 55, No. 1 (Jan., 1975), pp. 1-12 The University of Chicago Press
Shelley, Bruce L. Church History in Plain Language. 4th ed. Nashville, TN: Thomas Nelson, 2013.
Edit: Fixed the link to the AskBibleScholars sub.