50. Mishnah Berurah (584:1) writes, "Although we are certain we will be acquitted and found innocent in the judgment, one must be afraid due to the awe of the judgment. In the merit of his fear, he will be remembered [for a good year]."
Shulchan Aruch (597:1) states, "We eat, drink, and are happy, and we don't fast on Rosh Hashanah. However, one shouldn't eat until he is full so that he won't come to ראש קלות, lightheadedness on Rosh Hashanah, and he should have the fear of Hashem on his face."
The Rambam (Pirush HaMishnayos Rosh Hashanah 4) writes, "Hallel isn't recited on Rosh Hashanah and Yom Kippur because they are days of avodah, humility, fear, and awe of Hashem. These are days to escape and to run to Hashem, days for teshuvah, tefillos, requests, and forgiveness. Therefore, Hallel and joy aren’t appropriate on these days."
Notice that the Rambam refers to Rosh Hashanah and Yom Kippur as days of fear and days "to escape and run to Hashem." Unlike most fears that cause people to run away from them, the fear of Rosh Hashanah draws people closer to Hashem.
As we say in Selichos, בצילך מחמתך אתכסה, "I hide from Your anger in Your shadow." This is the uniqueness of the fear of Hashem. It draws us closer.
The Chasam Sofer taught another reason why we don’t say Hallel. He explains that it is known that our ancestors join us in our tefillos on Rosh Hashanah and Yom Kippur, and it is written, ה-י יהללו המתים לא, "the dead cannot say Hallel"; therefore, we don’t say Hallel on Rosh Hashanah and Yom Kippur.
The Meiri (Rosh Hashanah 16.) writes, "On Rosh Hashanah, the judgment is for the living and the dead, whether their future will be one of pleasure or distress, wealth or loss."
