A good question. Hopefully this post will satisfy my questioners.
Before I give my answer proper, observe that early Stoics resembled Cynics. Aristo of Chios is a good example: he rejected physics and logic entirely to focus exclusively on theoretical ethics. History records his comparison of dialectical reasonings to cobwebs--both look nice but are otherwise useless. Zeno of Citium, the first Stoic, also criticized dialectical reasoning. It is arguable that the Cynic impetus that began Stoicism provided the anti-logical opinion espoused by early Stoics. However, flares of anti-intellectual sentiment continued until the codification of Stoic doctrine by Chrysippus, the third scholarch. His elaboration of Zeno remade the Stoic school into a much more intellectual enterprise. Propositional logic found a place in his philosophy. Interestingly enough, however, the first person to define the term proposition was the proto-Cynic Antisthenes. I believe anomalies such as these reveal the versatility of Cynic and Stoic philosophers. Despite prevalence of anti-intellectualism in both schools, the characteristic non-conformity of these philosophers excuses their development of theoretical disciplines.
Having established an early similarity between the two schools, I now turn to the critical difference. Arguably it is not a substantial one, at least, not at first glance. Julian the Apostate said that the difference between Stoics and Cynics is in their manner of dress. Essentially, according to him, their doctrine is the same or very similar. A prominent characteristic of Ancient Cynicism is the shamelessness of its practitioners. Widely abused and rarely appreciated, shamelessness was the best criterion by which a Cynic could be distinguished from any other kind of Hellenistic philosopher. Contemporaries could not match the disregard with which Cynics treated the customs of their day. While unnecessarily ostentatious displays of shameless brought charges of exhibitionism against less discriminate Cynics, the overall purpose of the technique was pedagogical. By being shameless, Cynics hoped to attract an audience. Other motives can be imagined, but the function of Cynic as pedagogue holds the most appeal for me personally.Some ancient writers complain of the abuse of Cynic license. Unscrupulous people, sometimes even the illiterate, would don the uniform of the Cynic and use the title to commit acts of indecency. They would then, of course, expect immunity. False Cynicism was the greatest enemy of Ancient Cynicism. It is clear that rouges such as these cannot be judged representative of the philosophy. Cynics despised them more than anyone else. Barring from consideration the usurpers of Cynic parrhesia, I believe that most Cynics used shamelessness only when it benefited their followers or laypeople generally. Accounts are given in Diogenes Laertius that portray Zeno of Citium being made to walk down the busiest section of the Athenian marketplace holding a pot of lentils. This was extremely embarrassing for him, but Crates demanded it. Zeno tried to conceal the pot of beans, so Crates smashed it with his staff, causing the mush to dirty Zeno. Completely humiliated, Zeno ran away. Crates chased after him, claiming that nothing bad had happened. This is the best example of the Stoic and Cynic perceptions of shamelessness--the former considers it only a source of embarrassment, while the latter counts it a necessary part of character development and even public instruction.
This divarication is the core difference between Ancient Cynicism and Ancient Stoicism. While Cynics find shamelessness necessary for philosophy, Stoics disagree to the point of making appropriate social behavior a virtue. Acting inappropriately in social relations can even be as serious as defying Nature in the Stoic ethics. Suppose one has a bad father. The relationship of father to child, according to the Stoics, does not require us to consider whether the father is bad. It only requires the child to obey the father. Should the father be abusive or ungrateful, the child is still obligated to obey him. Epictetus makes his point clear: social relationships necessitate obedience. There is no room for philosophical displays of disagreement. To defy the natural order is unethical. No further consideration is necessary. I find this rigid, almost legalistic subservience to be oppressive. And the Stoics would argue that relating to a father is only one of many social capacities in which involvement is required. All of these demand obedience if one is to be considered ethical. Contrast Cynic ethics. Should a father be evil, and a child finds himself compelled to obey, it is to the shame of the father. Expose his evil, do not conceal it. Diogenes of Sinope was said to have hated evil people for their moral depravity and good people for their silence in the presence of evil people. We see here a powerful difference in the approach of the two schools to the same problem.
The most substantial consequence of Stoic appropriateness is that people are restricted in their enthusiasm for philosophy. One may be philosophical, but only so much. To defy Nature by behaving inappropriately is inherently unphilosophical. While I do not want to argue too much about the Stoic consideration for society, I suggest that this may be the underlying reason for the difference. The Cynic is unafraid to live without society; the Stoic values social order. If it is at all possible, the Stoic prefers to live within society. It is this preference that inhibits the Stoic from revealing social corruption. He or she is compelled to be a good citizen. Cynics recognize the superfluous nature of society, and this recognition frees them to be as enthusiastic as they wish. In fact, enthusiasm is required of the Cynic: he or she must be self motivating without the support of society.
But why would anyone wish to be without society? Would it not be better to simply renounce offensive and shameless behavior for the sake of enjoying the benefits of civilization? Not for the Cynics. Moral compromise to any extent is inexcusable. We are taught that society exists so long as people are willing to give up certain offensive aspects of their individuality for the sake of the collective. Should individuals decide to no longer have consideration for each other, society brakes down. We are free only so long as we do not infringe on the freedom of others--I believe that is the commonly accepted formulation. But the Cynic rejects this. He or she demands absolute freedom. Diogenes of Sinope compared himself to the mythical Hercules on the basis of their being alike in freedom. Knowing that one is self-sufficient empowers the Cynic to act shamelessly when necessary. Should society fail, the existence of the Cynic continues. Why should Cynics compromise when loss of freedom is the only result? They find nothing gained from society, so what do they have to loose?
Ultimately I believe this to be the deciding reason in my preference of Ancient Cynicism over Ancient Stoicism. While it is arguable that living in an anarchistic environment would not be pleasant, and the failure of society is by no means something to be hoped for, the ethical purity of the Cynics still outweighs the compromise of the Stoics. Antisthenes taught that society was useful, but not necessary. He compared it to a fire: get too far away and you freeze; too close, you burn. Perhaps we should live at a distance. Cynics live this virtue, as I hope to. While Stoics are admirable and a close second to the Cynics in the listing of my favorite schools, I will always prefer the shamelessness enthusiasm of the latter to the appropriate compromises of the former.