I have selected the Ayas from the Quran where these two words appear together.
Here you are:
When those come to thee who believe in Our signs, Say: "Peace be on you: Your Lord hath inscribed for Himself (the rule of) mercy: verily, if any of you did evil in ignorance, and thereafter repented, and amend (his conduct), lo! He is Oft- forgiving, Most Merciful. (Quran 6:54)
Between them shall be a veil, and on the heights will be men who would know every one by his marks: they will call out to the Companions of the Garden, "peace on you": they will not have entered, but they will have an assurance (thereof). (Quran 7:46)
This is a difficult passage, and Commentators have interpreted it in different ways. Three distinct schools of thought may be discerned in the interpretation. 1. One school thinks that the men on the Heights are angels, or such men of exalted spiritual dignity (e.g., the great prophets), as will be able to know the souls at sight as regards their real worth: the Heights will be their exalted stations, from which they will welcome the righteous with a salutation of peace, even before the righteous have entered heaven; the salutation of peace being itself an assurance of salvation to those whom they salute. 2. Another school of thought thinks that the men on the Heights are such souls as are not decidedly on the side of merit or decidedly on the side of sin, but evenly balanced on a partition between heaven and hell. Their case is yet to be decided, but their salutation to the righteous is a wistful salutation, because they hope for Allah's Mercy. 3. The third line of interpretation, with which I agree, is close to the first, with this exception, that the partition and the Heights are figurative. The higher souls will rejoice at the approaching salvation of the righteous.
"Peace unto you for that ye persevered in patience! Now how excellent is the final home!" (Quran 13:24)
(Namely) those whose lives the angels take in a state of purity, saying (to them), "Peace be on you; enter ye the Garden, because of (the good) which ye did (in the world)." (Quran 16:32)
And when they hear vain talk, they turn away therefrom and say: "To us our deeds, and to you yours; peace be to you: we seek not the ignorant." (Quran 28:55)
The righteous do not encourage idle talk or foolish arguments about things sacred. If they find themselves in some company in which such things are fashionable, they leave politely.
And those who feared their Lord will be led to the Garden in crowds: until behold, they arrive there; its gates will be opened; and its keepers will say: "Peace be upon you! well have ye done! enter ye here, to dwell therein." (Quran 39:73)
In the above verses, "peace be with you" is a greeting of those who practice faith. Not only a greeting to each other, but also the greeting of angels, those who standing on heights or living in celestial gardens. But peace is also to those who are ignorant and we wish they could find the right path.
Let's think a bit. In Hebrew, people greet each other in a similar way. Sholom Aleichem is also a sublime greeting in Judaism, a wish for peace. Interestingly, for the people of the Middle East, peace is an unattainable desire that comes pretty much only in the afterlife. I have already lived in some parts of the world where this greeting would be very strange. They would be puzzled and asked back: what's wrong with you? For us, peace is the state of natural existence! What the hell do you wish peace for us?
The Scriptures may also shed light on the fact that as we break away from our natural life forms and feel privileged within ourselves, the distance increases between us and others. That causes hatred, unrest, and we get farther and farther away from peace in our earthly existence, so peace remains an afterlife wish. Or don't you experience that peace is an empty phrase, but it means war in reality? Look, what organizations has a name where peace is mentioned? International Liaison Committee of Organizations for Peace (1946-1961), World Peace Council, peacekeeping forces and many more.
These were organizations that have eroded the true meaning of peace through their war or cold-war activities.
We Monotheists are at the forefront of earthly shit-mixing because we have not propagated in our history the expectations laid down in our faith but the perverted unrest of man. We shall fail when others learn our mentality and practice their supremacy on us the same way as we did with them. Today, words do not give their original context. Thus, with today's text-comprehension, it is very difficult to reproduce a content that has been revealed thousands of years ago, where peace, love, promise still had real meaning.
Whether there is a peace or not, intention and the respect for each other is the essence, which frankly speaking begins at the greeting, in that spirit:
When a (courteous) greeting is offered you, meet it with a greeting still more courteous, or (at least) of equal courtesy. Allah takes careful account of all things. (Quran 4:86)
We give kindness and courtesy without asking, and return it if possible in even better terms than we received, or at least in equally courteous terms. For we are all creatures of One God, and shall be brought together before Him.