The following is based on a "Yahrtzeit Shiur" delivered by Rav Soloveitchik.
In one of the Rav's *!97**) Shiurim, the Rav illuminated some of the main concepts dealing with Kiddush and Havdala. There are a number of conceptual questions which he attempted to answer with his various Chidushim on Kiddush and Havdala. Some of those questions follow: First, what is the purpose of Kiddush and Havdala? Why is there a separate Kiddush and Havdala first in Tefilla (see Berachot 33b) and then over a ,&2 (cup of wine)? Furthermore, if one were to recite Maariv early, would he be allowed to recite Kiddush in his Davening and which Havdala would he recite? A Halacha concerning Kiddush and Havdala is that one is forbidden to eat before it is recited. What is the reason for this, and if one were to accidently eat before reciting Havdala, is Kiddush or Havdala still recited? In terms of Kiddush, is the drinking of the wine an integral part of the Mitzva or is it simply the Beracha itself recited on the wine that matters? There is a Halacha concerning Havdala that if one forgets to make Havdala in his Davening, the person simply makes it separately within the Havdala over a ,&2. However, if someone forgets the Havdala in his Davening and over a ,&2 he must go back and recite it both in the Tefilla and over the ,&2. What is the reason for this? Finally, the Rambam (Hilchot Shabbat 29:17) holds that any (/9 /$*1%, national drink, including brandy, can be used when making Havdala. However, these same drinks are not permitted to be used for Kiddush. Only wine (or bread under certain circumstances) may be used. Why is there a difference in drinks for Havdala and Kiddush?
The Rav begins to answer these questions as follows. There is a separate Kiddush in Davening and 3- %,&2 because while both of their purposes are to separate Kodesh from Chol, they do it in different ways. There are :1* !258)*. (two aspects of the Mitzva) in regards to this separation. The Kiyum (fulfillment of the mitzva) in Tefilla applies to the Menucha aspect of Shabbat, or the Isur to do Melacha aspect of Shabbat. The Havdala on wine, on the other hand, separates or deals with the ,"&$ &3&1# aspects of Shabbat, or the eating aspect. This is the fundamental difference between the two.
The question of whether or not one could make Kiddush if he were to recite Maariv early (see Berachot 27) hinges on the above concept. It depends on one's definition of what ;&25; :"; (extending the duration of Shabbat beyond the basic halachic minimum) does. If Tosefet Shabbat deals with simply the prohibitions being extended, then only Kiddush in Tefilla can be said. However, if Tosefet Shabbat also extends to the Kedusha aspect of Shabbat then the Kiddush on the ,&2 can be said as well.
There is a Machloket in the Gemara (Pesachim 100) as to whether one must stop eating when Shabbat starts and before one makes Kiddush on a ,&2. Rebbi Yosi says that you do not stop at all and only make Kiddush after you finish the first meal, which is still a 23&$; (&- (a weekday meal). Rav Yehuda says that you must stop immediately and bench and then start again with Kiddush. He believes that the meal becomes a 23&$; :";, a meal of Shabbat, and for that, it must be restarted and Kiddush must be said. Shmuel believes that you do 5&92 /5% &/8$:, you interrupt your eating, say Kiddush, and continue the meal. The meal is converted into a Shabbat meal and that is okay, you do not need to recite Birkat Hamazon but do need to make Kiddush. The three disagree on whether or not a 23&$; (&- can be converted to a 23&$; :";. However, they all agree that it is not Shabbat that triggers the (*&" of Kiddush. Rather, it is the meal that triggers the (*&". The Kiddush's purpose is to be 8&"3, to establish, the 23&$; :";. The Kiddush characterizes the eating on Shabbat. This is the reason why it is forbidden to eat before making Kiddush. If one were to eat before making Kiddush, it would defeat the very purpose of Kiddush. This is the basis for the opinion that you do not make Kiddush if you do eat first (Pesachim 106-107). However, we hold that Kiddush is made in such a situation. This is also the reason why Kiddush can be made over bread if wine is unavailable. If the purpose of Kiddush is to be 8&"3 the meal, bread can certainly accomplish this. The Rambam says (Hilchot Shabbat 29:6), however, that it is a Rabbinic Mitzva to make Kiddush on wine, because, as the Rav explains, you have the added dimension of :*9% 3- %**0, song over wine (see Tosafot in Brachot 41b s.v. !*).
This is also why it is the wine itself that is a key to the Mitzva, not just the Beracha said over it, because it is to be 8&"3 the meal. However, the question can be raised that if it is the drinking of the wine which is the (*&", why can someone be /&7*! someone else? In the Seder on Pesach, each individual drinks the four cups of wine. With the !9"3 ,&2&;, one cannot be *&7! with someone else's Kiddush (see Tosafot in Pesachim 99b s.v. -!). However, there is a difference between the !9"3 ,&2&; and wine for Kiddush. By the !9"3 ,&2&;, the wine is the main part of the Mitzva. But by Kiddush, it is a combination of the words of Kiddush and the wine that is 8&"3 the meal.
It is understandable that Kiddush is 8&"3 a meal and therefore it is forbidden to eat before making Kiddush. However, why is it forbidden to eat before making Havdala? The same Machloket that arose concerning Kiddush arises with Havdala, between Rav and Shmuel. Shmuel says 5&92 /5% and Rav says that Shabbat does not establish the obligation of Havdala so one would not have to stop if he ate before making Havdala. Once again, it is clear that it is not Shabbat that triggers the obligation to make Havdala but the end of the meal or the end of eating. However, the ban on Melacha ends with Maariv. Just as Kiddush has a dual aspect to it, so too does Havdala. The Havdala in Davening parallels Kiddush in Davening and ends the Menucha of Shabbat. The Havdala on wine parallels the Kiddush on wine and it ends the ,"&$ &3&1#.
It can be asked, however, why on Yom Kippur, when we do not eat any food, is there still a Havdala afterwards? The answer is simple. The Mitzva of ,"&$ &3&1# does not apply merely to food. The ,"&$ part includes nice clothing, etc., so is apparent that Havdala must still be said.
It is clear from what we said concerning Havdala that Havdala is in fact connected to the meal and is in a sense, 8&"3 the meal because it signals the end of 23&$; :";. This is why it is forbidden to eat before Havdala and if one did eat, the purpose of Havdala is defeated and some say that Havdala is not said.
It is possible to take the Rav's concepts of Kiddush and Havdala and use them to answer why if one were to forget both Havdala in Davening and Havdala on wine he must recite both, whereas if one misses both Kiddush in Davening and on wine he only recites Kiddush on wine. While both Havdalot are used to separate different things, if one misses Havdala in Davening, the Havdala on the wine can still separate between Chol and Kodesh. However, once the Havdala on the wine is missed, its power is diminished to such an extreme that it cannot be used instead of Havdala in Davening, so both must therefore be repeated.
It is also possible to answer why the Rambam says that national drinks can be used for Havdala but not for Kiddush. Kiddush, or the beginning of the Shabbat meal, symbolizes the beginning of Kedusha and requires only Kodesh items to be used. However, Havdala symbolizes the end of Kedusha and the onset of Chol and therefore drinks of all kinds can be used.