The fundamental difference between Rosh Hashanah and Yom Kippur may be compared to the two ways in which one can go about destroying a building.  The first option is to tear down one layer at a time, one wall at a time, until the entire building has been dismantled.  Alternatively, one can place a stack of dynamite underneath the building and have it explode, collapsing the entire building in a single instant.

Rosh Hashanah is the day when we proclaim God's Kingship over the world.  This experience has the effect of destroying the evil within us in a single blow.  When we focus on God as the King over the entire universe, when we contemplate His unparalleled power and majesty, our natural, instinctive response is a desire to repent, to draw closer to God.  At once, our evil inclination and impulses are subdued and we are aroused to Teshuva.

On Yom Kippur, however, we follow the other method of "destruction," confessing each of our sins, one by one, correcting each of our spiritual flaws one at a time.  Unlike on Rosh Hashanah, on Yom Kippur we "dismantle" one layer of impurity after another, until, hopefully, the entire "building" is entirely destroyed.

Is it permissible to cry during the prayer service on Rosh Hashanah?  Given the status of Rosh Hashanah as a festive holiday, should we forbid crying, despite the sense of awe and trepidation one naturally experiences on this day?

Chacham Ovadia Yosef addresses this question and concludes that a person who is overcome with emotion during the Rosh Hashanah prayer service is not required to restrain himself from crying, but one may not endeavor to bring himself to tears on this holiday.  He draws proof to his ruling from the narrative in the Book of Nechemya (chapter 8), which tells of an assembly held on Rosh Hashanah for the Jews who returned from Babylonia to rebuild the Beit Ha'mikdash.  Ezra read to the people from the Torah, and upon realizing how far they had strayed from Torah observance, the people began to cry.  Ezra and Nechemya urged the people not to cry, and to instead rejoice in the festive spirit of the holiday.  This demonstrates that crying in inappropriate on Rosh Hashanah, and therefore one should not make an effort to bring himself to tears on this holiday.

The Arizal, however, was of the opinion that crying is to be encouraged on Rosh Hashanah, and indeed many authorities allowed crying during prayer on Rosh Hashanah.  How might we reconcile this view with the aforementioned narrative in the Book of Nechemya?

Some Rabbis suggested a distinction between the specific context of prayer and the rest of the day of Rosh Hashanah.  Crying is permissible on Rosh Hashanah as part the experience of Tefila, as one pours his heart before the Almighty.  But throughout the day, after one leaves the synagogue, it is inappropriate to feel despondent and anxious; on this day we are to joyfully celebrate God's kingship and feel confident in His infinite mercy and compassion.  Therefore, Ezra and Nechemya urged the Jews of their time to return to their homes and celebrate Rosh Hashanah with food and drink, rather than spend the entire day mired in distress and anguish.

In conclusion, a number of works mention that the inspiration and spiritual intensity we sometimes feel while we pray during the year originate from our impassioned prayers to God on Rosh Hashanah.  If a person prays with concentration and sincere emotion on Rosh Hashanah, the power of these prayers exerts influence throughout the coming year in the form of these moments of inspiration.  It therefore behooves us to recite our prayers on Rosh Hashanah with intense feeling and concentration, as this will help us gain inspiration during our prayer services throughout the coming year.