Why does a priest say "The Lord Be With You” and a bishop say "Peace Be With You" as a liturgical greeting?

Why does a priest say "The Lord Be With You” and a bishop say "Peace Be With You" as a liturgical greeting?

In a spiritual sense the bishop take the place of Christ as the head of his diocese. The he is has the privilege of saying: Peace be with you!

The main reason although not expressed as such is that the pope and all bishops are considered the successors of the Apostles (of Our Lord, Jesus Christ). Priests are normally under the jurisdiction of their local ordinaries (bishop).

Pax vobis (or vobiscum), like the other liturgical salutations (e.g. Dominus vobiscum), is of Scriptural origin. The Gospels contain such forms as: "veniet pax vestra, "pax vestra revertetur ad vos" (Matthew 10:13), "Pax huic domui" (Luke 10:5), "Pax vobis" (Luke 24:36; John 20:21, 26). The salutation, "Gratia vobis et pax" or "Gratia misericordia et pax", is the opening formula of most of the Epistles of St. Paul and of St. Peter, and occurs also in those of St. John as well as in the Apocalypse. The formula was quoted from the Old Testament by Our Lord and His Apostles (cf. especially "Pax vobiscum", "Pax tecum", Genesis 43:23; Judges 6:23), and was thus naturally preserved in the liturgy and in Christian epigraphy as a memorial of Apostolic times. Like the Dominus vobiscum, it was first used in the liturgy (in the form of Pax vobis) by the bishop in welcoming the faithful at the beginning of the Mass before the Collect or the Oratio. When the Confiteor, Introit, Gloria in excelsis were added at a later period, the Pax vobis or the Dominus vobiscum was preserved. The form Pax vobis is now employed by bishops and prelates only — Dominus vobiscum being used by priests — at the first Collect. Hence the Dominus vobiscum became the ordinary introduction to all the orations and most of the prayers. The Greeks have preserved the Pax omnibus or Pax vobiscum. There was formally a certain rivalry between the two formulae, Pax vobis and Dominus vobiscum, and some councils (notably that of Braga in 563) ordained that both bishops and priests should employ the same form of salutation (for the texts, see the bibliography). Besides this episcopal or sacerdotal salutation, the words Pax tecum, Pax vobis, or Pax vobiscum are used in the Liturgy at the kiss of peace. On such occasions the Liturgy contains prayers or collects ad pacem (cf. Kiss; Cabrol in "Dict. d'archéol. et de liturgie", s.v. "Baiser de Paix", where all references are given). In the Ambrosian Liturgy, at the end of the Mass, the people are dismissed with the words: "Ite in pace" (cf. "Auctarium Solesmense", 95). Dom Martene (op. cit. in bibliography, III, 171, 174) gives other instances of the use of the word Pax. In Christian epigraphy there is a variety of formulae: pax; in pace; pax tecum; vivas in pace; requiescat in pace; pax Christi tecum sit; anima dulcissima requiescas in pace; dormit in pace; in locum refrigerii, lucis et pacis (from the formula in the Mass at the Momento of the Dead). - Pax in the Liturgy (Catholic Encyclopeia)

Prelates who are priests (abbots) may also say the phrase ”Peace be with you" because they enjoy many privileges reserved to bishops. For example, benedictine monasteries are under the jurisdiction of their abbot and not the local bishop. The abbot takes the place of Christ in the monastery, just as the bishop does in his diocese.

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