Misesians vs. Hayekians
That is an interesting question. For a more in-depth analysis I recommend Murray Rothbard's essay "The Present State of Austrian economics" where he analyses the difference between the Misesian version (which Rothbard belonged to) and the Hayekian version. Before I describe the difference between Mises and Hayek I should point out that there were actually "two" Hayek's, just like there were two versions of Hayek's cousin Ludwig Wittgenstein. Early in his career, Hayek was a Misesian and argued for the Austrian business cycle theory during the 1920s and 1930s. It was on the basis of this that Hayek received the Nobel price in economics in 1974. But then after World War II, he lost interest in business cycle theory and instead started to argue for the ideas described below.
In short, the Misesian version is based on what Ludwig von Mises called praxeology, which means the logic of action. Praxeology is based on a number of axioms, the most important of whom are the action axiom: the human beings act on purpose using means to achieve ends. Bases on this and a number of related axioms and laws of reality, you can then deduce a number of implications. Among the more immediate ones are that people will always choose the action they believe ex ante (before the action) will produce the best results, otherwise they wouldn't have acted that way. A second implication is that since the future is uncertain, people can't be certain that the action they perform will turn out to be the best. This uncertainty also implies that people will have to make forecasts about the future when deciding on which means to use in order to achieve the ends they want. But as people can only choose one specific form of action, every action is associated with a opportunity cost in the form of the actions they chose away.
Misesian praxeology then continues to deduce new logical implications in the above manner and apply them to more specific situations with certain specific conditions, such as the existence of money controlled by a central bank, advanced division of labor and a heterogeneous capital structure, and from that Misesians can derive advanced theories such as the Austrian business cycle theory. This approach, which is known as the axiomatic-deductive approach, is similar to the one in Mathematics and formal logic and is really no less certain than it. While Misesians can be certain of the economic laws implied by praxeological reasoning, they can't be certain of how applicable certain theories will be, but then again this is no different from how you can't be certain of how applicable mathematical conclusions like 103+201 are or for that matter how applicable physical laws and conclusions like how often airplanes will actually fly.
Misesian praxeology is thus based on logical analysis of how people try their best to use ends to achieve means and what that will mean given specific conditions.
Hayekians by contrast are not really interested in this kind of approach. For them reason is not an ally for acting man, it is a dangerous illusion and an enemy. The point of the Hayekian analysis is that acting individuals really can't alone come to any correct conclusions. Instead, truths and rules gradually evolve in a collective process governed by the anonymous spontaneous social forces, and while no one can really come up with anything individually, they can still learn somehow from the spontaneous social evolution, an evolution which is similar to the unplanned evolution we see in nature.
The Hayekian argument against socialism and government intervention consists in that socialism is based on the illusion that reasoning by any individuals (in this case politicians and other government officials) can be used to improve conditions. What Hayekians fail to see is that this argument will apply not only to government decisions but also to decisions made in any organization, including private companies. Indeed, it could even be made against any decision-making by individuals that doesn't involve simply mindlessly following what the group somehow comes up with collectively. By contrast, the Misesian case against government intervention consists in using reason in the form of praxeological reasoning to demonstrate the negative effects of such interventions.
In short, the difference between Misesians and Hayekians is a radically different view of reason and logic. To Misesians, people may be fallible, but they can and should still use reason to analyze the world around them and use these conclusions to improve the odds that the course of action they choose will turn out to be the best. To Hayekians, by contrast, belief in reason is a "fatal conceit" as Hayek puts it and the basis of socialism, while mindlessly following rules and ideas spontaneously evolved by the group is somehow the basis of a free society.
Again, I recommend you to read Rothbard's longer analysis of this subject.