I'll defer to a dictionary for a "standard" definition, but as I understand these two terms...
Heresy is adherence to a belief which is similar in some respects to a biblically-sound belief, but which differs in some key respect, placing it outside the realm of orthodoxy. Usually this word is reserved for beliefs which deny core doctrines of the faith. For example, Arianism denies the full divinity of Jesus Christ... and is considered a heresy by most Christians. The word is sometimes overused and applied to difference of belief within Christianity.
Apostasy is "falling away" from belief. One who accepted the teaching of Christianity, and began walking with the Church eventually rejects Christ for one reason or another and begins to teach against the faith. This is different from mere backsliding, in which a Christian steps away from active faith for a time and may return later (please, let's not argue OSAS here!). Modern examples of apostates are Bart Ehrman and Charles Templeton.
Wikipedia has decent articles on both issues.