The Shorter OED doesn't give adaption, as @ig0774 says. It'll be in the Complete, though, because it's definitely a valid word. But I certainly agree with others, adaptation is to be preferred.
Exactly the same has happened with adoption and adoptation, except in that case the shorter word became the standard form.
So far as I'm concerned these are (rare) examples of true synonyms. Unsurprising, maybe, since they use very similar (and both legitimate) ways of deriving a noun from a verb. But if we're to consider the smallest of details, it's fair to say that in each case, the less-common form has overtones of long-winded verbosity.