שים שלום
A difference exists between giving (נתינה) and placing (שימה). The word נתינה indicates extending from one place to another known place in whatever fashion. Whereas, שימה indicates placement with direct intention into a prepared and guarded place.
The source of the difference in words is based on the Gemara (Temurah 34a) regarding the verse in the matter of the removal of the ashes from where piled on the Altar, ושמו אצל המזבח. The Gemara explains that the concept was ושמו, with ease, ושמו, and not spread out, which means that it should be placed easily and not with quickness and not spread out.
On the other hand there is another Gemara (Gitin 77a), regarding one who throws a Get to his wife. When the wife is in the middle of the house, or in the middle of a courtyard or into the bowl in front of her, in such cases the wife is divorced. The Gemara asks why she was divorced even though the Get was not actually placed into her hands, only into her possession (since the Torah says into her hand בידו). The Gemara learns this is effective since the verse says ונתן, in whatever manner the Get was transferred to the wife’s possession was effective.
The intention of this Gemara was that if the Torah meant to limit the placement of the Get only into her hands, then the verse would have used the word שימה and not used the word נתן. The word יד includes the concept of possession (like the use of the word in Parshas VaYeshev ויצברו בר תחת יד פרעה) which would be limited by שימה or expanded by the word נתן.
Now we can explain the words שים שלום. Shalom is a general concept which can indicate a non-specific peace (lack of fighting), spread out and not effective on a specific place (like the indications of the word נתן). However, the peace that we all seek and pray for, is a specific peace that exists not from a lack of fighting, but exists directly between people and in specific places (like the indications of the word שימה).
