So I know my homoiousios vs. homoousios, and my monophysite vs. dyophysite, and my monothelite vs. dyothelite, and how it all led to the Arab caliphates getting a decent navy and winning the Battle of the Masts.
I don't, and I'd love to! (If you feel like it, obviously.) I'm pretty sure the homoiousios one is about, like, the Trinity or something, but beyond that it's all Greek to me.
(At this point, I feel like I owe @apocrypals royalties or something, but I'm getting a weird kick from doing this on Saint Patrick's Day, so let's do this).
I covered the impact of the monophysite vs. dyophysite split and the Battle of the Masts here, so I'll start from the top.
You are quite correct that the homoiousios vs. homoousios split was, like most of the heresies of the early Church, a Cristological controversy over the nature of Christ and the Trinity. This is perhaps better known as the Arian Heresy, and it's arguably the great-granddaddy of all heresies.
The Arian heresy was the subject of the very first Council of the early Church, the Council of Nicea, convoked by Emperor Constantine the Great in order to end all disputes within the Church forever. (Clearly this worked out well.) In part because the Church hadn't really sat down and attempted to establish orthodoxy before, this debate got very heated. Famously, at one point the future Saint Nicholas supposedly punched Presbyter Arius in the face.
What got a room of men devoted to the "Prince of Peace" heated to the point of physical violence was that Arius argued that, while Christ was the son of God and thus clearly divine, because he was created by God the Father and thus came after the Father, he couldn't be of the same essence (homoousios) as the Father, but rather of similar essence (homoiousios). Eustathias of Antioch and Alexander of Alexandria took the opposing position, which got formulated into the Nicean Creed. As this might suggest, Arius lost both the debate and the succeeding vote that followed, as roughly 298 of 300 bishops attending signed onto the Creed. This got very bad for Arius indeed, because Emperor Constantine enforced the new policy by ordering his writings burned, and Arius and two of his supporters were exiled to Illyricum. Game over, right?
But something odd happened: the dispute kept going, as new followers of Arius popped up and showed themselves to be much better at the Byzantine knife-fighting of Church politics. About ten years later, the ever-unpredictable Constantine turned against Athanasius of Alexandria (who had been Alexander's campaign manager, in essence) and banished him for intruiging against Arius, while Arius was allowed to return to the church (this time in Jerusalem) - although this turned out to be mostly a symbolic victory as Arius died on the journey and didn't live to see his readmission.
....and then it turned out that Constantine the Great's son Constantius II was an Arian and he reversed policy completely, adopting the Arian position and exiling anyone who disagreed with him, up to and including Pope Liberius. While the Niceans eventually triumphed during the reign of Theodosius the Great, Arianism unexpectedly became a major geopolitical issue within the Empire.
See, both during their exile and during their brief period of ascendancy within the Church, one of the major projects of the Arians was to send out missionaries into the west to preach their version of Christianity. Unexpectedly, Arianism proved to be a big hit among the formerly pagan Goths (thanks in no small part to the missionary Ulfilas translating the Bible into Gothic), who were perhaps more familiar with pantheons in which patriarchal gods were considered senior to their sons.
While they weren't particularly given to persecuting Niceans in the West, the Ostrogothic, Visigothic, Burgundian, and Vandal Kings weren't about to let themselves be pushed around by some Roman prick in Constantinople either - which added an interesting religious component to Justinian's attempt to reconquer the West.
