Antisocial behavior can be characterized as an overall lack of adherence to the social mores, and standards that allow members of society to coexist peaceably. Many people who display this type of behavior may seem charming, but often cause harm to others, and show little remorse for their actions. Antisocial behavior can be part of a larger conduct disorder, or personality disorder, such as antisocial personality disorder, and it is said to be seen in men more commonly than women.
On the other hand, an asocial individual is someone who keeps to themselves, generally ignoring other people. Just as asexual is without sex, asocial is without socialization. These asocial people tend to be on the autistic spectrum. They are perfectly healthy, they'd rather not socialize, and frankly, judging from the behavior of a lot of students at in school, I can't say I blame them. The asocial individual does not commit antisocial behavior. They just ignore everybody. Someone who doesn't enjoy socializing at parties might be described as either "asocial" or "antisocial", but "asocial" is too mild a term to describe someone who commits an antisocial act such as planting a bomb. "Asocial" suggests indifference to or separation from society, whereas "antisocial" more often suggests active hostility toward society.
So, which one are you?