Adaption and adaptation are different forms of the same word, and they share all their meanings, which include (1) the act of changing to suit new conditions, and (2) a work of art recast in a new form or medium. But the longer word, adaptation, is preferred by most publications and is much more common. Adaption is not completely absent, but it usually gives way to the longer form in edited writing.
Both forms are old. The OED lists examples of adaption from as long ago as the early 17th century. Adaptation is just a little older, having come to English from French in the middle 16th century. Adaption has never been the preferred form, though, and in fact has grown less common relative to adaptation over the centuries.
It’s possible that some English speakers now view adaptation and adaption as separate words each with their own uses, but any such emerging differentiation is not yet borne out in general usage. For now, at least, adaption always bears replacement with the more common form.
This ngram, which graphs the use of adaption and adaptation in a large number English-language books and magazines published in the 19th and 20th centuries, shows that there is no contest: