Let’s start with the most important area of disagreement. The Stoics were out-and-out materialists. Everything in the world is made up of the four elements, either in pure form out in the cosmos or mingled in the things of the world. Even soul is made of pneuma, a kind of heated air. The universe is bounded and outside it there is void, nothing, no thing at all. The material cosmos is all there is.
Plato, however, was a dualist. He held that there were both material and immaterial things. He was led to this conclusion by reflecting on Socrates’s search for definitions. Since nothing material, nothing in the world, is securely what it is, then these material things cannot be the objects of definition. The things of this world come into existence and perish, increase and diminish. Anything we think of as beautiful in this world may be ugly in another respect, or at another time, or in another context. So when we try to define beauty, Plato believed, we are actually picking on something that is not of this world. When we see something beautiful, we are reminded of the existence of this immaterial thing, the Form or Idea of Beauty. In his dialogue Phaedrus, Plato describes the realm where Forms exist as literally beyond the rim of heaven.
And he also held that our souls are immaterial, because it is with our souls that we get to know Forms and we could not do so unless soul was just as immaterial as them. For the Stoics, soul was refined matter, and they argued that if the soul was not material it could not have an effect in the material world. If the soul was not material, they said, it could not impel our material body to act. To exist is to act and be acted upon. Actually, that’s an idea they took from Plato as well. They borrowed some such incidental ideas from Plato – for instance, that a good analogy for the soul’s reception of impressions from the outside world is to think of marks being impressed on a wax block. But these are only incidental.
So that is the major and fundamental difference between Platonism and Stoicism. But of course the Stoics learned something from all the major schools of thought that preceded them: the Presocratic philosopher-scientists, Plato and his successors in the Academy, Aristotle and his school. In fact, Plato’s influence grew over the centuries of Stoicism’s existence, which is to say that later Stoics were more prepared to compromise the austere philosophy of the earliest Stoics, Zeno, Cleanthes, and Chrysippus.
Stoic cosmology is a good place to start, because it shows areas of agreement and disagreement. The disagreement, as you’d expect, was that Stoics rejected Plato’s transcendent creator god. Their god was immanent within nature, and was identified with nature, in its purposive, creative, and providential aspect. For the Stoics, God was the rational intellect of nature. The Stoics took over from Plato the idea of the cosmos as a living being created by a rational god, but whereas Plato’s god, and more especially that of his successor Xenocrates of Chalcedon, looked to those immaterial Forms to do the work of creation, the Stoic God looks only to what is best for his creatures, and especially humankind. But another thing they took from Plato is the idea that evil is a necessary concomitant of good; the cosmos is bound to contain both at once. And even the famous Stoic idea that few things are good or bad, and most things indifferent, is probably based on a distinction Plato made (in Gorgias).
Socrates’s influence on the Stoics was far greater – or at least the influence of ideas that Plato and Xenophon, our two main sources, attributed to Socrates. Above all, many Stoics saw Socrates as a perfected sage, an example for us to emulate; so naturally they took over certain of his ideas.
First, Xenophon has Socrates clearly lay out the argument from design – the argument that God must exist because the universe is programmed, so there must be a programmer. The Stoics took that over lock, stock, and barrel.
Second, they believed in the so-called Socratic paradox, that virtue is knowledge, and they defined each of the virtues as a kind of knowledge, so that courage, for instance, is knowledge of what is and what isn’t truly fearful. And if virtue is something that only a person of knowledge can have, then to have one virtue is to have them all, which is also an idea Plato floated (in Protagoras). But the really important aspect of the paradox for the Stoics was this: if virtue is knowledge, then vice is ignorance. So they agreed with Socrates that no one does wrong deliberately. Everyone, even a serial killer, believes that what he’s doing is in his best interests; it’s just that he’s mistaken, because he’s destroying his soul. The only harm we can take is what we do to ourselves.
Third, Socrates spoke of philosophy as care of the soul, and it turns out that caring for one’s soul is being a virtuous person. The Stoics seized on that as well: they saw philosophy as therapy and as a way to progress towards virtuous sagehood. Like Socrates (especially Xenophon’s Socrates), they saw freedom as a product of personal self-discipline.
Fourth, in Phaedo, Plato’s brilliant reconstruction of Socrates’s last day on earth, he has Socrates describe philosophy as preparing for death, and fear of death was one of the primary feelings the Stoics (the Epicureans too) were concerned to combat.
Fifth and finally, a more entertaining point. A particular influence on Marcus Aurelius in his Meditations was Socrates’s assertion, in Xenophon at any rate, that we don’t need knowledge of physics and so on to make progress towards enlightenment, Marcus seized on that with a sigh of relief because he knew that he wasn’t very good at those abstruse subjects!