Just a little historical aside.
It’s only in Rome where the bishop got to have a different form of greeting:
Pax vobis(cum) “Peace be with you” rather than
Dominus vobiscum. (In Spain, there was even a local council (the second Synod of Braga in 563) which ordered that all the bishops and priests in Spain stick to
Dominus vobiscum - any other greeting was seen as smacking of Priscillianist heresy.) It is claimed that
Pax vobis arrived to Rome from the East (where it is the default greeting in the liturgy) during the 4th-5th centuries.
The reason for the difference may be connected to the
Gloria in excelsis Deo (the “Glory to God in the highest”). Originally in Rome, the
Gloria was only used in an episcopal/Papal Mass (originally only on Christmas; later extended to Sundays and feasts of martyrs). It was only much later that priests were granted the privilege of having the
Gloria at their masses by degrees. (At first, they were allowed to use it only on two occasions: their ordination mass and Easter Sunday or Easter Eve - which led at least one contemporary author to complain: why aren’t priests allowed to use it on Christmas as well, where it would have been more appropriate?)
Apparently there was an early sentiment that “Peace be with you” was a greeting suited for joyful/solemn occasions - and because the
Gloria was used on such occasions, the greeting was used
only for masses where the “Glory to God” was sung. In other words, only during a mass celebrated by a bishop. Because priests (at that time) did not use the Gloria at their masses, they usually only said “The Lord be with you.”* (In fact, in masses where there was no
Gloria the bishop would also use
Dominus vobiscum rather than
Pax vobis.) Even after the priests were finally allowed to use the
Gloria, the custom stuck.
- I should note however that this was apparently not a hard-and-fast rule yet during the early 9th century (800s): a priest may theoretically use either form. But we do know that a century later, during the time of Pope Leo VII (936-939) the connection between the “Peace be with you” greeting and the Gloria is already made.