The New American Standard Bible was translated from the Greek (and Hebrew for the Old Testament), while consulting Aramaic texts, so the Vulgate doesn't really come into consideration. Here's the Greek:
Ἰωάννης μαρτυρεῖ περὶ αὐτοῦ, καὶ κέκραγεν λέγων, Οὗτος ἦν ὃν εἴπον, Ὁ
ὀπίσω μου ἐρχόμενος ἔμπροσθέν μου γέγονεν· ὅτι πρῶτός μου ἦν.
The word in question is ἔμπροσθεν, which can be translated as before, in the sense of having priority in time or place. However, it can also have the sense of priority with respect to rank.
I consulted Vincent's Word Studies for this particular verse, and he said the following:
Is preferred before me (ἔμπροσθέν μου γέγονεν) — Literally,
"is become," so Rev., "or is here (compare John 6:25) before me."
Before is used of time, not of dignity or rank. The expression is
enigmatical in form: "my successor is my predecessor." The idea of the
superior dignity of Christ is not a necessary inference from His
coming after John, as, on that interpretation, the words would imply.
On the contrary, the herald who precedes is inferior in dignity to the
Prince whom he announces.
J. B. Lightfoot holds that this verse carries a similar idea to that of Philippians 2:6, in which Christ is exalted (ὑπερύψωσεν) to the highest place.
The word πρῶτός also means before in phrase "He existed before me", therefore the context seems to demand that some difference be made between these two words which can both mean before, so that the latter can serve as an explanation for the former. It appears that the translator felt that the distinction probably hinged on John's attributing a greater dignity to Christ.
