Yeah, Nestorius made Christ out to be suffering from multiple personality disorder. Big problem that needed to be addressed.
That’s probably not true. Nestorius denies that he teaches two sons. He says that there is one Christ with two natures, which of course is orthodox.
There are disagreements today about what the real issues were. Part of this is because most of his works were destroyed. The largest one survives only in translation. Also, it appears that his section of the church used terms differently.
As far as I can tell, the underlying issue is anhypostasia, the idea that there's no human person in Christ. This is the normal version of orthodox Christology. When I read Athanasius, the impression I get is that there’s no Jesus. The Logos wields a human body, but there’s no real human being. Indeed one gets the impression from Athanasius that there isn’t a human mind, though he never quite says it. When that issue was taken up later, it was decided that there actually was a human mind. Later on it was decided that there was a human will. But still, there isn’t exactly a human person. It may be an exaggeration to say that the Logos assumes all the parts of a person without an actual person, but I often get that impression.
My layman’s impression of Nestorius is that for him the Logos assumed not just human nature but an actual human being. However it’s still an Incarnation, there’s still just one Son of God.
Both views have their disadvantages. But it’s not obvious that we need to consider either non-Christian.
It may be that this is in part terminology. Nestorius was operating within a different tradition, and used a term “prosopon,” which isn’t exactly the same as hypostasis. So it’s not entirely clear that he actually said there were two hypostases in Christ. A prosopon is a bit more individualized than a nature, but probably not as much as a hypostasis.
Acquinas’ explanation of the Incarnation deals with this issue by saying that Christ’s human nature is individualized. That is, the Logos assumes not just all the parts of a human, but an actual individual of human nature. So why isn’t he Nestorian? Why isn’t he saying that the Logos assumes a human person? Because there’s a special rule that prevents calling Christ’s human individual a person, even though it’s identical in nature to every other human person.
For him a hypostasis is a complete, independent entity. Christ’s human nature isn’t complete, because it’s part of Christ. So even though it’s exactly the same thing as any other human person, we don’t call it a person because it’s the incarnation of the Logos. It seems to me that this may be what Nestorius was getting at by saying that it while the Logos assumed a prosopon, there’s still only one Son. It is quite possible that Nestorius’ prosopon is Acquinas’ individualized human.
Acquinas’ treatment is clearly better, but I’m not sure that Nestorius was trying to say was actually heretical.
I think it’s pretty obvious from context that the initial problem with Nestorius was that he denied that Mary was the Mother of God. That led Cyril to interpret him as unsympathetically as possible.