They both taught heresy in that they either thought God could change as in the Orientals, or that Jesus Christ was not really God as in the Nestorians. Plus they do not have the great saints like Maximus the Confessor, John of Damascus, Gregory Palamas, and others. To be fair, I did not spend much time trying to find writings from their saints after they left the Church that speak on their defense of these views of the nature of Christ. I am also very uninformed about the history of saints like St. Isaac the Syrian and others that are claimed to be Nestorians, I have read his writings and love them, but they don’t discuss this topic of division. I did read Theodoret of Cyrus and St. Cyril of Alexandria’s debates on the topic and clearly St. Cyril makes the most sense against Nestorianism. I know that Theodoret was not a Nestorian, but he did try to make sense of them as not being completely wrong, which he later had to renounce. As for mono-physites, I don’t exactly know why they thought that Christ could not have two natures and thought it best to split from the Church.
Why do you ask?