Rosh Hashanah and Yom Kippur tend to be lumped together as the "Yamim Noraim", with really long prayers, somber tunes, repentance, and so on. Perhaps surprisingly, they in fact have very different basic themes. To give a general, simplified overview:
- Rosh Hashanah's purpose is to commemorate the fact that God is king of the universe. This is necessary background for the holidays which follow later in the month. The shofar has two main historical purposes: as a declaration of solemnity (like at a king's coronation), and as a kind of siren in wartime, connoting a sense of urgency and crisis. Both are appropriate for Rosh Hashanah. Historical allusions in the prayers (the Patriarchs, Noach, and Mount Sinai) are less relevant to Jewish history than they are to creating a sense of God's authority, power, and awe over the whole world. We request, mainly, that these qualities be made obvious and recognized by all mankind.
- Yom Kippur begins with Israel's sin regarding the golden calf. Atonement for this sin was achieved through Moshe's prayer, and more formally through special sacrifices in the newly built Mishkan. In later history, these are paralleled by communal repentance and the Temple service. Thus, our prayers focus on confession, on forgiveness based on God's special relationship with the Jewish people, and on (restoration of) the Temple service. God's holiness is emphasized more than God's power.
As expected, the Rosh Hashanah theme is easily recognized in many RH prayers, and likewise for YK. In addition, however, there are also several prayers which are recited on both holidays, but at no other time of year. Some of them combine both holidays' themes in very interesting ways.
In particular, consider the "Uvechen" prayer which is inserted into the third blessing of every RH and YK Shemoneh Esreh. I will quote successive lines of it in italics, each followed by an explanation of which holiday the line "belongs" to.
1. "In every generation ascribe kingship to God; For he alone is exalted and holy."
A brief introduction to the whole prayer. It does not express what we hope for in the future, but what we should do now, in relation to the hopes which will next be expressed. The mention of kingship links it strongly to the RH theme, while holiness links it to YK.
2. "And thus, may your name be sanctified - Hashem our God - over Israel your people, over Jerusalem your city, over Zion the home of your glory, over the kingdom of the house of David your anointed one, and over your sanctuary and Temple."
A pure YK prayer. It calls for God to redeem the Jewish people and restore the Temple, and it mentions sanctity. There is no universalism or mention of power or kingship, ideas which the RH theme emphasizes.
3. "And thus, place your fear - Hashem our God - over all your handiwork, and your terror over all that you created. Let all beings see you, and all creations bow down to you; let them form one association to perform your will wholeheartedly. For as we know - Hashem our God - dominion is yours, strength is in your left hand and power in your right, and your name is awesome over all that you created."
A pure RH prayer. God's power over all mankind is repeatedly mentioned. The special relationship with Israel is not.
4. "And thus - Hashem - grant honor to your people, praise to those who fear you, hope to those who seek you, fluency to those who beseech you, happiness to your land, joy to your city, fulfilled destiny to David your servant, and resumed light to the son of Yishai, your anointed one - soon, in our days."
A pure YK prayer. Not only does it repeatedly mention the fulfillment of the God/Israel relationship, but it emphasizes the prayers of those who are repenting and calling out to God.
5. "And thus, the righteous will see and rejoice; the upright will revel; the devout will call out in delight - for corruption will disappear, and all evil will evaporate like smoke, when the reign of wickedness is removed from the earth."
The RH theme dominates here. The focus is the removal of evil from the entire world. The righteous people repeatedly mentioned could be non-Jewish. Even if they are specifically Jews, that fact is not considered important.
6. "And may you rule - Hashem our God - soon, alone, over all your creations; on Mount Zion the home of your glory, and in Jerusalem your holy city."
This line, the climax of the prayer, combines the RH and YK themes. One one hand, God will reign over the entire world. On the other hand, the Jewish people will be redeemed and returned to the Temple in Jerusalem. By suggesting that God will rule - but rule "in" Jerusalem - the goals of the two holidays are fused together.
7. "As it is written, 'Hashem will rule forever; your God, Zion, for all generations. Halleluyah.' "
We conclude with a Biblical verse to reinforce the requests. Here too, God rules - but specifically from Jerusalem. Thus, as in the final request, the two holidays' themes are united.
This particular prayer I see as an "anthem" of the Yamim Noraim. It expresses the themes of each holiday, alternating between one and the other. Lines 2 and 4 relate to Yom Kippur; 3 and 5 relate to Rosh Hashanah. In the remaining lines - 1, 6, 7 - the goals of the two holidays are united. God will be recognized by all. And Israel will repent and be restored. These goals are distinct, and each has a separate holiday devoted to it. But in the final analysis, in the view of Judaism, they are inseparable.