Gilson claims that it was the nature of universals that they could not agree on.
For Aristotle, universals existed virtually in things, needing the active intellect to draw them out. So universals were in the intellect, and they did not subsist as Plato had held (according to Aristotle.) Aquinas agreed with Aristotle with one caveat, he claimed that the trancendentals (being, goodness, unity, etc.) subsisted in the Divine mind. (cf. the proemium to his Commentary on the Divine Names)
Abelard, who was prior to Aquinas, was a logician who attempted to answer the question of whether universals subsisted. His teacher, William of Clairveux, was of a Platonic opinion, that universals subsisted. Abelard could not accept this because he saw that if we said that universals exist entirely in particulars, then in no sense are they universal. For example, if the essence (universal) ‘man’ was entirely in Socrates, then it could not be in Plato also, and then would not be a universal. If it was in each of them only in part, then there would be no commonality that could be called universal in them. Abelard settled the question in his own mind, though confusing to everyone else (in my and Gilson’s opinion), by stating that universals subsist as examplars in the divine mind, but that human beings only have some confused notion of them.
Ockham came along and said that universals do not exist at all, even in the divine mind. His thinking was along the lines of, whenever I have a universal thought, say of ‘man’, that thought is only a particular thought. Furthermore, he was in some sense a precursor to the skepticism of Hume, because in making all particular (nominalism), he assigned to God the creative activity of all causality. That is, all effects, essentially, he assigned to the creative activity of God. So when I strike a match, there is no effecient causality that starts the flame, it just happens to be the creative activity of God making that match spark. As such, all causality becomes the mere witnessing of a sequence of events, and there is no intrinsic efficient causality to nature. Hence, skepticism followed, as Ockham settled all philosophical questions by recourse to faith, i.e.God does everything (he even said we cannot know by philosophy that there is an immortal soul, but took recourse to the articles of faith.)
Bonaventure, on the other hand, assigned knowledge of the universals to the divine light. That is, anytime we come to know universals it is by infusion by the divine light.
I am being very general here, as there were a number of disputes over Aristotle in Aquinas’ own lifetime, but the essential grasp of why Aristotle was abandoned was the “problem of universals.” There were, for example, the Latin Averroists (see Siger of Brabant) who were considered ‘radical Aristotelians.’
If you read Aristotle’s *De Anima *he makes a claim that seems as though it denies personal immortality. That is, he says that the active intellect survives the body, but that the passive intellect dies with the body. So, the Averroists claimed that there was no personal immortality, which was obviously contrary to the faith. Aquinas settled this questions, in my opinion, by holding Aristotle to his principles more closely than Aristotle did himself, because Aristotle understood the intellect by analogy to sensation. Aquinas comes along and says that, if the radical Aristotelians are right, and the agent intellect is “somewhere out there” (Averroes actually held that it was on the moon), that merely means that the individual intellect is understood, not that it understands. (For more details, read Summa Theologica I. q.76 I believe.)
There were also disputes about the eternity of the world. In a nutshell, Bonaventure held that the eternity of the world was impossible because it was contrary to creation. The Averroeists held that the world was eternal, and Aquinas held that philosophy could not prove it one way or the other, but that we had to rely on faith to let us know that the world was created.
These are just some of the controversies, and this is a very general overview. As to why modern philosophers called themselves the “true heirs of Plato” I’m not really sure what philosophers you are referring to? This could be because there were disputes among the Franciscans and Dominicans, with the Franciscans following Augustine’s Platonism and the Domincans following Aquinas’ Aristotelianism. If you could name some philosophers who made this claim, I’d like to look into it.
Thanks, and I hope this helps. I really recommend reading Gilson’s book, as that is where I am primarily getting my information from. (Although I have read Bonaventure on the eternity of the world, as well as Aquinas.)