What are the main differences between moral and political philosophy? Is political philosophy an application of moral philosophy to the political structure of society?
There's a lot of overlap between the domains of moral philosophy and political philosophy. Consequently, there's not much historical differentiation -- Kant, Mill, and Aristotle both engage in political and moral philosophy.
The distinctions are in terms of the main questions that concern contemporary philosophers in each field of philosophy.
Moral philosophers are concerned with questions of right and wrong or questions of virtue. This can often be done without respect to the political structures that exist or seeing these only as a backdrop.
One common type of scenario in contemporary analytic moral philosophy is the trolley problem which involves whether to divert a trolley from killing more people to killing less people, etc. No one I'm familiar with asks whether you're living in North Korea, China, or France to figure out which would be moral. Some moral philosophers approach problems from the perspective of some moral framework or with reference to a historical philosopher like Kant. Others look directly at moral intuitions by using surveys.
Separately, some moral philosophers look at the emotions. For instance, we can see this in the work of Martha Nussbaum or by looking at Mencius' four moral roots (See http://plato.stanford.edu/entries/emotion/#10).
Conversely, political philosophers look at questions about how the state should be organized. This strain historically begins in Plato (Republic, Laws) and Aristotle (Politics and one part of Nicomachean Ethics). It keeps happening in Hobbes (Leviathan), Locke (Letters concerning Toleration, Two Treatises on Government), Rousseau (The Social Contract), Kant (Perpetual Peace, Religion), Hegel (Philosophy of Right, Natural Law, Hegel's System of Ethical Life) - just to name a few works.
To highlight the distinctions a bit more clearly, political philosophy is more apt to consider right/wrong action in terms of crime than morality (though not exclusively) whereas moral philosophy does not need to concern itself (in general) with whether something wrong is criminal.
A second aspect that has arisen more recently for political philosophy is problems of collective action and collective responsibility. Again, the distinction is not going to be absolute but when you shift to how things work in groups you're generally moving into political philosophy rather than moral philosophy.
To return to your specific phrasing,
Is political philosophy an application of moral philosophy to the political structure of society?
I would say no -- or at least no always. In general, there's a theory of how society should be organized that doesn't arise directly from moral principles. For instance, in Aristotle's account, the virtues do not scale up to create an ideal society. Instead, individuals are seen as part of larger organic wholes to which they belong (thus Aristotle declares that the polis precedes the individual and the family). Similarly, Mencius's political philosophy is built on an idea of society that is organic rather than on the individual.
There are of course views of society like social contract theories or Kant's Kingdom of Ends where there's deep integration between the idea of morality and political organization. For social contract theorists, an (imagined or hypotheszied) state of nature is overcome and transformed into a political realm where morality can also exist. For Kant, political society is society that contains rational beings who act in accordance with the categorical imperative and achieve harmony.
Maybe to sum all of this up, moral and political philosophy often differ in focus rather than content. A moral philosopher might look at the problem of drug addiction from the perspective of the incontinent will whereas a political philosophy might look at the problems with social organization that permit or deny the addict access to drugs and whether/what measures are acceptable to restrict this.
Moral philosophy is about what is right or wrong for an individual person to do, sometimes within the context of a larger moral structure attributed to the universe. Social/political philosophy is about the role, functions, virtues and desirable organizational structures of groups of people. Because both are about what is right and good for people to do, the same philosophers are often interested in both, but they can neither be characterized as always the same nor as always differentiable across the work of various philosophers.
In practice, it depends on the philosopher. Plato's Republic arguably introduces its political theories entirely in the service of explaining a moral theory (and both are an extension of Plato's metaphysics). For Plato, human society is has a fractal structure, the political system is just the moral system at a higher level of scale. In contrast, for Aristotle, each level of scale has its own characteristic virtues, and therefore the moral system and political systems are related but distinct.
For Sartre, the individual is primary, and the group can only be considered as the aggregate of the decisions of a free individuals. So individual morality is the foundation, politics is merely a function of the individual's absolute responsibility for his or her own world.