A common question posed by a certain Bais Din to potential converts is "What is the difference between the Sabbath and a holiday?" Alternatively, for those who only understand the Yeshivishe lingo, "What's the nafka mina between Shabbos and Yuntiff?" The food is the same (two breads, fish, soup, chicken/beef, some side dish, etc…) and the clothes are the same, yet every Jew knows that there is some inherent variation between Shabbat and Yom Tov. Where do Rosh Hashanah and Yom Kippur fall in? Candles are lit for all of these days, and nobody goes to work. However, are there any differences between these various holy days?
On Shabbos there are at least seven people "called up" to the Torah, while on a Holiday, the minimum is only five (or six on Yom Kippur). From this, we see that Shabbos is on higher spiritual level than its weekday counterpart is. While certain changes to the prayer for Shabbos are also done for the weekday holydays, there are certain prayers recited exclusively on Shabbos (i.e. qel Adon, yequm purqan, and parts of Qobalas Shabbos). These prayers, while theoretically could be fitted into the Yom Tov liturgy, are not, because of the higher spiritual energy emanating from Shabbos.
While Yomim Tovim do have their own special prayers, most of those are recited even on Shabbos (e.g. the Mussaf `Amida of Yom Tov, Ya'ele V'yav`o, and Yizkor when applicable) because they are explicit or implicit references to the festival itself which does not necessarily contradict the spirit of Shabbos. Those festival prayers that are skipped on Shabbos are inherently festival prayers and are not the in the Shabbos spirit because they mention certain ideas that are not to be discussed on Shabbos (i.e. Hosh`anos on Succos, the supplication recited following the Thirteen Attributes, and in some communities qoh kaylee).
Shabbos, described as a day of rest, is meant for us to imitate an action -- or more precisely perhaps inaction-of HaQodosh Baurch Hu (the holy One, blessed be He). Among other Mitzvos (commandments), Shabbos is a fulfillment of the commandment to cleave to HaShem's ways[1]. HaShem rested on the seventh day of creation, so too his nation/children/servants/flock rests on the seventh day of the week. The Festivals are also an imitation of something: They serve to remind the nation about our previous ventures. We do certain actions to arouse ourselves to think about events that occurred to us in previous generation. We eat Matzos on Pesach and sit in Succos on Sukkos to remind ourselves of our forefathers leaving Mitzrayim (Egypt). We light a Menorah and read the Megilla, the scroll of Esther, to remind us of triumph over persecution in the times of old. We fast at certain points of the year to mourn the destruction of the Bais HaMiqdash (the Holy Temple) in Yerushalayim (Jerusalem). Shabbos is an imitation of an act of G-d, while the other Yomim Tovim are simulations of what people did. Which is on a higher level?
The pre-Shabbos attitude differs from the attitude before a Yom Tov. Shabbos is seen as the "light at the end of the week." Shabbos is a day to rest from the (hectic) workweek. As we say[2], "six days shall you do work, and on the seventh day, Shabbas Shabbason qodesh la'shem." Shabbos is called a "Shabbos Shabbason," with the double usage of the word rest, as it is specifically sanctified as a day to HaShem. Yom Tov, on the other hand, is called both an "Atzeres" (lit. stoppage [of work]) for HaShem[3] and an "Atzeres" for the Jewish people[4]. It is not necessarily a halting of work purely for HaShem, but there is some personal gain behind it. The Mitzvah of Yomtov is "chatzi la'shem v'chatzi lachem" - half for HaShem and half for you. We are supposed to get our own enjoyment from Yuntiff. Shabbos is a holy day for davening and learning Torah.
Although the rarity of Yom Tov causes people to hold it in higher esteem than the weekly Shabbos, the opposite should be true. Since Shabbos is a weekly event, the individual has a closer connection to Shabbos and can anticipate it at any time (i.e. it is appropriate -even encouraged-- on Sunday to start preparing and awaiting the next Shabbos). This is in stark contrast to Yom Tov, which cannot be expected at any time (i.e. in the middle of November it is ridiculous to begin waiting for the next Yom Tov, Pesah, which could occur in almost half a year). Preparations for Shabbos run from one week to the next, but the preparations for Yom Tov last longer. People prepare mentally, physically, spiritually, and financially (e.g. you had better start saving up for that new dress that your wife wants for Yuntiff) for the Yomim Tovim well in advance. (In fact, this segment of the essay is being written in the middle of the summer.)
Not that there is a competition, but clearly, Shabbos is much "holier" than Yom Tov is. HaShem and the Rabbonim gave us more prohibitions for Shabbos than they did for Yom Tov. On Yom Tov, one is allowed to carry in a public domain even without an `Eiruv and to cook Yom Tov food using a pre-existing flame. On Shabbos, obviously, both of these are prohibited. This relates back to the idea that Yom Tov is for our enjoyment and therefore we are allowed to perform certain actions for our own enjoyment.
There are four days in the year known as the "Yomim nor`aim," or the "Days of Awesomeness:" The two days of Rosh Hashana, Yom Kippur, and Hosh`ana Rabah. On these days, the communal leaders (i.e. the Chazon, Ba'alei Kriy`ah, etc.), as well as the married congregants, wear a special garment known as a "Kittel." The Kittel is actually a white burial shroud, which serves to remind individuals about the awesomeness of HaShem, and his everlastingness, in addition to man's mortality. The "parokhes" (curtain covering the entrance to the Ark containing the Torah Scrolls) is customarily switched to a temporary white one, to remind ourselves to clean ourselves of sin (which is represented by red). Changes in prayer include the styling of HaShem as "the Holy King" in place of "the Holy G-d," as well as the description of HaShem being "awesome" (nor`a) after Shema (itself an affirmation of one's acceptance of the yoke of heaven) during the Torah taking-out service. On Rosh Hashana, one blessing in the Mussaf service is dedicated to the declaration of HaShem's kingship.
Rosh Hashana is noticeably more similar to Yom Tov than it is to Shabbos. Rosh HaShanah requires only five Aliyot, like any other Yom Tov. Prohibitions that are permitted on Yom Tov are likewise permitted on the Jewish New Years. Like other Yomim Tovim, Rosh Hashana is an annual occurrence and its presence requires the recitation of the "Shehechyyanu" benediction. Rosh Hashana comes with its own series of halachik rules and Minhagim (customs). One law is to hear the somber blowing of the Shofar (horn, preferably of a ram). Similar to the holiday-specific actions performed on other festivals, the main purpose of sounding the Shofar is to emulate and remember the actions of our ancestors. Avraham, our forefather, after having been told that he will not be sacrificing his only beloved son, Yitzchak, offered a ram on the alter on the mountain which eventually became the site of the Beis Hamikdash. One horn of this ram was used by Moshe Rabenu as a Shofer; and the other is hidden for future use by Moshiach. The Shoyfar was also instrumental in the Jews' battle for Yericho (Jericho). Comparable to other Biblically ordained rules about eating which apply during the other Yuntiffs, Rosh Hashana has customs about various foods that people eat as a siman (omen) for a good year.
However, Rosh Hashana is not actually one of the festivals. It is really a unique day of its own. Rosh Hashana has a unique property that is not shared by any other of the 383 potential days of the Jewish calendar: The Mussaf Amidah (Shemonah `Esrei) of Rosh Hashana is comprised of nine blessings. Additionally, Rosh Hashanah is not one of the three times[5] when every Jew is divinely decreed to visit the Holy Temple. Nonetheless, historically speaking, one holy Jew, Elkana Ben Yerucham, the father of Shmuel HaNavi (Samuel the prophet), did make a yearly pilgrimage to the Temple for Rosh Hashanah.
The sanctity of Yom Kippur seems to fall in between Shabbos and Yom Tov, but is really a category of its own also. Yom Kippur requires neither seven aliyahs like Shabbos, nor five like a Yom Tov, but rather six. While Yom Kippur is like Shabbos inasmuch as cooking and carrying are prohibited, it is also like Yom Tov because of the similarities in the prayers (i.e. Ya'ele V'yav`o in the Amida, and, if chas v'shalom one is sick and must eat it said in Bircas HaMazon, Yizkor, and outside of Eretz Yisro`el, the priestly blessings). Nonetheless, Yom Kippur is similar to neither for there are five times of prayer rather than the usual four of a Sabbath of Festival.
Although the recitation of the Thirteen Attributes of Mercy (Middos HaRachamim) on Yom Kippur is akin to the festivals, the recitation of Aveinu Malkeinu is distinctive to Yom Kippur. Moreover, this prayer even supersedes Shabbos for one-time a year in a certain rare instance. It is recited when Yom Kippur occurs on Shabbos only in the concluding prayer known as ne`ilah (the locking, a reference to the locking of the gates to heaven at the conclusion of Yom Kippur) when it is only deemed Shabbos due to a safeq (doubt). (It is also said at a time of an extreme and urgent emergency in which we need to request mercy. Perhaps these two instances are really one in the same due to the urgency of Yom Kippur.)
Yom Kippur is called a "Shabbos Shabbason" (with the double usage of the word rest)[6], as it is sanctified as a day to HaShem, just as Shabbos is. However unlike Shabbos, about which the Prophet says[7] "V'Qar`asa La'Shabbos `Oneg" ("You shall call Shabbos an enjoyment"), Yom Kippur is day when one is obligated to afflict his soul[8]. This affliction is achieved in five ways: By refraining from food and drink, by refraining from wearing leather, by refraining from marital relations, by refraining from smearing or anointing one's body, and by refraining from washing one's self. Such affliction on Shabbos is totally forbidden. On Yom Kippur, this affliction reminds us of our past misdeeds to stimulate feelings of regret, which commences the T'shuva (repentance) process.
All the holidays and holydays are similar in one major aspect. They are all a time for self-reflection, for one to make an internal accounting of one’s soul (Chesbon HaNefesh). A Jew's "holiday season" differs very much from that of the Gentile's season. While all their holidays are characterized by eating and drinking in a gluttonous fashion, a Jew is forbidden to do such behavior[9]. Each holiday has its own special Mitzvos, customs, prayers, and attitudes. Surely, one would not approach Purim with the same degree of formality as he would approach Yom Kippur nor would one dance around the Bima on Rosh Hashana as he does on Simchas Torah. However, do you see any difference between Memorial Day and Labor Day? Even on a Yom Tov which, as established above, is given for our own pleasure, we are constantly busying ourselves in Mitzvos for HaShem. The raison d'etre of Shabbos and Yom Tov is to continue building ourselves in the world through Torah, good deeds, and serving HaShem.
[1] Deuteronomy 10:20
[2] Exodus 31:15, conf. Leviticus 23:3
[3] Deuteronomy 16:8
[4] Numbers 29:35
[5] Deuteronomy 16:16
[6] Leviticus 23:32
[7] Isaiah 58:13
[8] Numbers 29:7, Leviticus. 23:24
[9] See Ramban on Leviticus 19:2