I am a student of Greek philosophy in two ways -- I both study the schools of thought academically and practice them in my life, especially with regard to Epicureanism and Stoicism. The two philosophies have been in my experience pitted against once another, painted as competing philosophies. My academic study of the two philosophies is very shallow: I have read from both philosophies’ texts, but I have never emerged myself in an in-depth study of how they were received in the Greek world at the time, so I don’t know how accurate such a portrayal is. On the surface, it would seem to make sense. The Stoics believed that virtue was the only “Good”, and the Epicureans believed that happiness was the only “Good”.
Do they contradict one another? Both begin from absolute statements that on the surface differ from the other, and one philosophy is grounded in divinity while the other is not. Epicurus had little regard for the metaphysical: he believed that happiness in the here and now was what people should focus their attention on. The ancient Stoics believed in cosmic order and saw this Order as the source for all that is good -- like beauty, truth, and virtue. To live in compliance with this Order is to live with virtue and thus be happy. The chief Stoic doctrine is to “live according to Nature”: living within our limits. Epictetus, whose work I enjoy immensely, began his Handbook by stating that happiness can be achieved through the knowledge that there are some things we can control and some things we cannot . To act on this knowledge is to live with virtue. But notice what Epictetus is focusing on: happiness. This is the same then Epicurus was focused on.
This is why I do not think that Stoicism and Epicureanism are actually contradictory. Each seem to begin with the object of human happiness as their goal, they just attempt to reach it through different (but not necessarily opposing) practices. Epicurus advocated the simple life and abstaining from insatiable pleasures: Stoics believed in mental discipline, the cultivation of mindfulness. But what would stop Epicures from using Stoic mindfulness, and what would stop Epictetus from living the simple life? The two philosophies differ only in theoretical beginnings, I think, and for the modern Stoic or Epicurean, that simply doesn’t matter.
I count myself a Stoic, but I do not believe in a living Cosmic Order the way Marcus Aurelius and Epictetus did. Erik Wiegardt commented in his The Stoic Handbook that the difference between an atheistic Stoic and a pantheistic Stoic is that one believes cosmic order is unconscious and the other believes it is conscious. I believe laws govern the universe, but I do not think they are divine. I believe in gravitation and friction and inertia and thermodynamics and all manner of universal laws, but I think they are natural. What lies beyond them -- what caused them -- is not my concern. I can no more be aware of supposed metaphysical realms and gods than can a microbe be aware of a soda can.
If I take Epicurus’ approach that supposed metaphysical worlds are meaningless when it comes to human happiness, on what basis do I call myself a Stoic? I do so because there are certain patterns of behavior that lend themselves toward happiness and unhappiness. If I become addicted to a drug, for instance, I will be on the whole unhappy. This is not divine punishment being meted out by Athena: this is chemistry. If I fret about what someone is thinking of me, I will be unhappy. Again, there are no punitive deities involved: this is psychology. If, however, I becoming addicted to substances and adopt the Stoic practice of giving no attention to things I cannot control, I will find contentment -- and the joy I have for living will not be tainted. “Virtue” is the practice of living sensibly, by following patterns of behavior that create long-term happiness. For me, Epicureanism and Stoicism go well together, because the virtuous life is -- in Epicurus’ own words -- the happy life. 
 My immediate source for this is Sharon Lebell’s The Art of Living, but the same sentiment is expressed in the same basic way in more conservative translations of Epictetus' works.
 “The Principle Doctrines”, Epicurus