A contraction is a word formed by two or more words put together and omitting some letters. (can't = cannot, won't = will not)
An abbreviation is a shortened form of a word or phrase. It is not considered a legitimate word itself; a contraction, on the other hand, is indeed considered a complete word. (km = kilometer, med. = medical)
ASAP is not an abbreviation, by the way. It is an acronym - a word made up of the first letters of several words.