[per Webster's New World dict.] Both advent and adventure are derived from the same Latin infinitive of advenire ('ad- v/wen- ear- ay': to come), but the two may have been differentiated in meaning right from the very beginning. The term Advent for the observance of weeks leading up to Christmas is taken from the past participle adventus (has come); while on the other hand the dictionary notes a (hypothetical) vulgar Latin concurrent usage of the word adventura, which it translates as "a happening".
The verb advenire was then incorporated into old French directly (dropping the final 'e') as advenir (to come about/ occur), while the noun's concept of exciting involvement was somewhat distinguished by dropping the 'd' to get aventure (an experience/ ). The import of aventure was still close to that of advenir in that culture though, in the sense that ~.
Then, when French was melded into middle English, the word "venture" was adopted: either by aphesis of just dropping the initial 'a' or perhaps as hearing the word as "a` venture" (at a venture). And venture came to have an interpretation more in opposition to "a coming upon", as more like outwardly running a risk/ placing a stake, and venturing out/ abroad. However, venture (-someness) still implied a guardedness, tentativeness or hesistancy of activity which didn't sit well with those who wished to be thought adventurous, so ~.