Here's a bit of a problem with regular utilitarianism (or any kind of consequentialism that deals with mental states like happiness). You've got to act in such a way that you maximise the happiness of those around you, and they've got to act in such a way that they maximise your happiness. Weirdly, the responsibility for your happiness seems to fall on everybody else, and you bear a fragment of responsibility for everybody else's happiness. This is contrary to the way we intuitively think about happiness - we think that our own happiness is to a decent extent our own responsibility. Be satisfied with what you've got, we say, and so on. But what makes ourselves happy is often very difficult to pin down - we can come into a load of money, feel oddly unsatisfied, and then experience nirvana while watching that evening's sunset. How much more difficult is it to fathom the causes of and take responsibility for other people's happiness?
The Stoic answer to this problem has the advantage of being simple. Happiness, it says, is entirely our own responsibility. Nothing anybody else can do ought to make you happy - if it does, you're behaving immorally and laying yourself open to inconsistency and danger. Make your happiness depend on your own virtue, and you will be entirely self-sufficient, since nobody but you can assist or harm your virtue.
What, then, is our responsibility to others? We should act in a gentle, dignified, kind way, but that's a function of our responsibility to ourselves, not to other people. In terms of other people's benefit, Socrates' last wish was that the Athenian state should punish his sons harshly and thus lead them to virtue. This is obviously directly at odds with a utilitarian view.
Does a Stoic have a responsibility to others at all? Maybe not.