According to Random House Unabridged Dictionary, the similarity between the two is superficial. Adult comes from the Latin word adultus, which is the past participle of adolere (to make grow) Adultery can be traced back to the Latin verb adulterare (to mix, to adulterate)
The root of the former verb is -ol, and the root of the latter one is -ulter.
This similarity is, I think, what is referred to as a "false friendship". The etymology of the two words IS different: ad + alescere is Latin for "to grow up", and the past form is adultus. adulterare is a Latin verb meaning to debauch, corrupt, probably itself derived from ad alterum (-am etc) se conferre meaning "to betake oneself to another".
At least, so says my Oxford Dictionary of English Etymology.