What's the difference between sim shalom and shalom rav? Why replace sim shalom at mincha and maariv?
Kipa.co.il links שים שָׁלום to the priestly blessing according to Orach Chayim 127 (2), so whenever there is a possibility for the priestly blessing, שים שלום is said. The trouble there is that there is no priestly blessing to Shabbos mincha and there is a minhag to say שים שלום (and another to say שָׁלום רָב). Kipa.co.il links the שים שלום minhag on Shabbos mincha to the mention of תּורַת חַיִּים in שים שלום.
(Text of both below)
I have heard the explanation the other way round too. שים שָׁלום contains the words תּורַת חַיִּים and became the choice for shacharis and all other occasions when there is a reading from the Torah (e.g. a fast day at mincha).
שָׁלום רָב does not contain a reference to Torah and is used when there is no reading from the Torah.
The only snag is that some versions of Nusach Ashkenaz say שים שָׁלום on Shabbos mincha even though there is a reading from the Torah.
שים שָׁלום טובָה וּבְרָכָה. חֵן וָחֶסֶד וְרַחֲמִים עָלֵינוּ וְעַל כָּל יִשרָאֵל עַמֶּךָ. בָּרְכֵנוּ אָבִינוּ כֻּלָּנוּ כְּאֶחָד בְּאור פָּנֶיךָ. כִּי בְאור פָּנֶיךָ נָתַתָּ לָּנוּ ה' אֱלהֵינוּ תּורַת חַיִּים וְאַהֲבַת חֶסֶד. וּצְדָקָה וּבְרָכָה וְרַחֲמִים וְחַיִּים וְשָׁלום. וְטוב בְּעֵינֶיךָ לְבָרֵךְ אֶת כָּל עַמְּךָ יִשרָאֵל בְּכָל עֵת וּבְכָל שָׁעָה בִּשְׁלומֶךָ:
שָׁלום רָב עַל יִשרָאֵל עַמְּךָ תָּשים לְעולָם כִּי אַתָּה הוּא מֶלֶךְ אָדון לְכָל הַשָּׁלום וְטוב יִהְיֶה בְּעֵינֶיךָ לְבָרְכֵנוּ וּלְבָרֵךְ אֶת כָּל עַמְּךָ יִשרָאֵל בְּכָל עֵת וּבְכָל שָׁעָה בִּשְׁלומֶךָ.
According to http://www.aishdas.org/asp/shalom-rav it is partly based on Nusach Bavel vs Nusach Eretz Yisrael
Two variants of Shalom Rav were found in the Cairo Geniza. Given the similarities between Cairo and Israeli nusachos, it is assumed to have been the Nusach Eretz Yisrael in the days of the tannaim and Israeli amora’im. Another indication is that Nusach Provence, the berakhah was always “Shalom Rav“. The Provencial community, which was a true center of Jewish Life and Torah in the days of the rishonim, had a strong tradition of being refugees from the Roman conquest of Israel. In fact, the Jews of Luneil had a tradition that they were descendants of the city of Yericho. The name “Luneil” is from “luna” (French: moon) in the same way “Yericho” would be from “yareiach” (Hebrew: moon). Whether or not this tradition is historically true, it does reflect the community’s roots in Israel. So it would seem Shalom Rav was the Nusach Eretz Yisrael.
This is one of a number of cases where Sepharadim follow Nusach Bavel and Ashkenazim find a compromise between Nusach Bavel and Nusach Eretz Yisrael. Among many historians, these cases are caused by the number of captives from Eretz Yisrael taken back to Italy by the Roman army, whose descendants then go up to Ashkenaz in Charlemagne’s day.
The Provencal community ended up joining Ashkenaz. The bulk of Ashkenazic Jewry, though, were invited up from Italy by Charlemagne, who needed an economic backbone for the heart of the Holy Roman Empire. But many of these Italian Jews were also descendents of Israel; many taken back to Rome with the departing armies.
Another example is the preservation of the Israeli nusach by saying it in only some situations is “LeDor vaDor“, using it instead of “Atah Qadosh” when saying Qedushah. In Israel (and the Cairo Geniza), LeDor VaDor was simply the text of the third berakhah in every Shemoneh Esrei.
So it looks like splitting our time between Sim Shalom when this is a tefillah in which duchaning is allowed and Shalom Rav when it is not is a legacy of how Minhag Ashkenaz was formed. Descendants of Israel and Bavel relocated to the same area, and had to make a common minhag for this community, and so the result was a bit of compromise.
This idea about the origins of Minhag and Nusach Ashkenaz (including these two cases, piyut, and much more) are discussed at length in the writings of Prof.s Agus and Ta-Shma, and others.