Was there a particular reason? Did his Christian upbringing never sink into him? I've heard that St. Basil or another church father who knew him personally growing up found him immensely proud and vain, and that this was the reason for his rejection of Christian belief. But did Julian ever write out a specific reason?
There is surely someone who can answer this question more in depth, but let me give you a small, provisional answer.
Julian the Apostate (331) was born just a few years before Constantinine's death in 337. After his death his Arian sons wanted to solidify their own positions as heirs to the empire, and they decided to kill almost all of their relatives, including roughly all of Julian's close family members. He was subsequently brought up in a Christian way, exiled for a while and welcomed back again. During this time, he at least showed some interest in biblical studies, as is evident from his later writings.
He studied neoplatonism, at the time a mixture of pagan and christian thought, from 351 onward. According to his own writings (362), he was a Christian for twenty years and a pagan for twelve. If his own account is to be believed, he might have decided to study neoplatonism because of his paganism, not the other way around. The logical conclusion in this case would be that he apostated because of the trauma's from his youth, although this is extremely hard to prove.
On the other hand, he also befriended Gregory of Nazianus and Basil the Great in 354. These Christian men seem like an odd companion to a pagan who was traumatized by Christians from his childhood onward. However, during this time he also inaugurated in the Eleusian mysteries, a pagan cult which was held in high regard by pagans for centuries. This would mean that his own writings were at least true to some extent, since from roughly the middle of the 50's onward we can regard Julian as an apostate.
Why he apostated is thus highly difficult to say, since (as far as is known to me) he did not write about the exact reason himself. It is most likely a combination of his aversion to Christianity because of the events in his childhood and his rather pagan upbringing at the neoplatonist school.
On the other hand, it must also be mentioned that it could have been a power-move: the empire can be considered to have been in disarray compared to Constantine the Great's time, and the previous emperors were all Christians. It is clear that Julian had a completely different view of what a Good Emperor should be like compared to his predecessors, and he idealized, among others, Marcus Aurelius Reign. Portraying himself as a pagan could have been a smart PR-move to make sure that the people saw him as one of the Emperors of Old, and thus would accept him more. This would be in line with the fact that he did not prosecute Christians, but rather retracted their advantages over the pagans. In this case, it would have been a very pragmatic move to restore the empire to it's previous (pagan) glory again, without making the mistake of strengthening the Christians through persecutions. However, this does not line up neatly with his own writings who are not only in favor of paganism but also ridicule Christians, thereby undoing part of the pragmatism which would be embedded in his shift to paganism.