There are several holidays that we mark, commemorate, and observe during the course of the year, and we are now entering a month of concentrated Yamim Tovim, starting with Rosh Hashana. In most cases, the Torah states the significance of the day(s), and Chazal and later rabbis expand. Pesach celebrates liberation; Sukkot celebrates harvest and our survival in the desert. Yom Kippur is the day on which our nation receives atonement (see Vayikra 16:17). Shavuot, on the level of what the Torah writes (which is somewhat limited) is the time of reaping in Eretz Yisrael, and there are observances that have to do with the corresponding service in the Beit Mikdash (see Shemot 23:16 and Vayikra 22:17-22). Chazal teach us (as is somewhat clear from thep’sukim) that this is the period of the giving of the Torah.
What about Rosh Hashana? The Torah says nothing specific about the day’s significance, other than listing several special korbanot (fewer than any other yearly holiday) and the fact that there are some "horn blasts" or related remembrance. Chazal, of course, uncover the great significance. Rosh Hashana commemorates the creation of the world, and it is the day on which all of mankind is judged on a yearly basis (Mishna, Rosh Hashana 1:2). Yet, if we think about it again, we should wonder if there is any national element to this important day. All of mankind was created, and all of mankind is judged! Yet, we are the only ones who are commanded to commemorate the day. Do we or do we not have a special connection to the day?
One of the best ways of answering the question is by looking at the special berachot of Musaf of Rosh Hashana, which is a text that is paralleled only by the Haggada of Pesach in providing a framework for understanding the day. The first beracha, Malchuyot, is quite universalistic in talking about the need for the whole world to "coronate" Hashem in the broadest sense. Yet, it also discusses how a prominentAm Yisrael is a tool to bring this about. The beracha concludes "… King over all the land, Who sanctifies Israel and the Day of Remembrance." The second beracha, Zichronot, is also a mix of the universalistic, invoking the story of Noach, and national, discussing Avraham’s binding of Yitzchak and the resulting covenant. It ends off "… He Who remembers the covenant". The third beracha, Shofarot, is the one which clearly focuses on Israel, as we are the ones who communicate with Hashem with the shofar, starting with Hashem using it at Sinai and with our praising Him with it. The beracha concludes "… Who hears the sound of the blasts of His nation, Israel, with mercy."
We summarize as follows. Rosh Hashana is important for the whole world. However, the opportunity and obligation to act on it by means of a specific set of commemorations, with the shofar standing above all others, is reserved for us. Since the other nations do not have that, they are less likely to gain significantly from the day, which is balanced by the fact but they are not responsible or culpable for failing to do so. "Fortunate is the nation which knows the blasts."