It's also pronounced as /adˈweː.nit/ when it's the third person perfect tense of venio, with the accent on the e.
They're both grammatical but they're not interchangeable. In neither case is ādvenit an accurate spelling, the initial-a is short whether it's stressed or unstressed.
advēnit is the third person perfect tense, and advenit (ádvenit) third person present. I had no idea about the distinct accentuation. Thanks for pointing that out.
I translated vespers as 'in the night' - I believe it was used for that before on duo. no?