In the context of liturgy (in the beginning of Mass), only the bishop is allowed to say “Peace be with you” in the beginning of Mass. As to why it is so, we can speculate that it’s likely because the bishop has the fullness of the priesthood, and thus being so like Christ in his office, he is thus authorized to say it. I have yet to pore into the rubrics to be sure.
As far as scripture goes, the salutation is from Ruth 2:4 and 2 Chronicles 15:2 in the Vulgate.
In Ruth, the phrase appears in the sentence, "Et ecce ipse veniebat de Bethlehem dixitque messoribus: 'Dominus vobiscum'. Qui responderunt ei: 'Benedicat tibi Dominus'." ("[Boaz himself] came from Bethlehem and said to the harvesters, 'The Lord be with you!' and they replied, 'The Lord bless you!'").
II Chronicles recounts that Azariah said, "Audite me, Asa et omnis Iuda et Beniamin! Dominus vobiscum, quia fuistis cum eo. Si quaesieritis eum, invenietur a vobis; si autem dereliqueritis eum, derelinquet vos." ("Hear me, Asa and all Judah and Benjamin! The LORD is with you when you are with him, and if you seek him he will be present to you; but if you abandon him, he will abandon you.")
The phrase additionally appears in Numbers 14:42: "Nolite ascendere: non enim est Dominus vobiscum: ne corruatis coram inimicis vestris.”
It is also a clear instruction from the Liturgiam authenticam (De usu linguarum popularium in libris liturgiae Romanae edendis) dated 28 March 2001. The document instructs the requirement that translations of the liturgical texts from the official Latin originals, or Sacred Scripture from the original Hebrew, Aramaic, and Greek must be:
Insofar as possible, must be translated integrally and in the most exact manner, without omissions or additions in terms of their content, and without paraphrases or glosses.
More into speculating why only bishops are authorized to say Pax Vobiscum:
Pax vobis (or pax vobiscum) are the words Our Lord used in John 20:21 to give peace to his apostles and to send them out to give His peace to others. It seems initially to have been used only by the Pope, a usage transfer to the bishops in the middle ages, as more or more areas of the West adopted the "pontifical" Mass. The Missal of bishops is called the Pontifical Missal to this day. This greeting seems appropriate to the Pope and the bishops, as the successors of the apostles.
Pax Vobiscum reserved to bishops