Gemara

(Mishnah): When Yom Tov is after Shabbos, we make Havdalah (in Kidush of Yom Tov). The text of this Havdalah is 'ha'Mavdil (He who separates) Bein Kodesh l'Kodesh';

(R. Zeira): When Yom Tov falls in the middle of the week, the text of Havdalah is 'He who separates between Kodesh and profane, and between light and darkness, and between Yisrael and the Nochrim, and between Shabbos and the six days of work.'

Pesachim 53b (Rav Yehudah): We bless on fire only on Motza'ei Shabbos, for that is when it was created.

54a: Ula and Rabah disagree. They hold like R. Binyamin bar Yefes;

(R. Binyamin bar Yefes citing R. Yochanan): We bless on fire on Motza'ei Shabbos and on Motza'ei Yom Kipur.

102b: Rav taught that when Yom Tov is on Motza'ei Shabbos, one blesses on the wine, Kidush, light and Havdalah (in this order).

104a (Beraisa): One says Havdalos on Motza'ei Shabbos, Motza'ei Yom Tov or Motza'ei Yom Kipur, even when Motza'ei Shabbos is Yom Tov or Motza'ei Yom Tov is Chol ha'Mo'ed, but not when Motza'ei Yom Tov is Shabbos.

Berachos 29a (Rav Nachman citing Shmuel): (There is an abridged version of Shemoneh Esre in which the 13 middle Berachos are condensed into one Berachah (Havinenu).) One may pray Havinenu any time of year, except for Motza'ei Shabbos or Motza'ei Yom Tov, for then one must say Havdalah in Chonen ha'Da'as (the fourth blessing, for understanding).

Question (Mar Zutra): One could make Havdalah part of the long middle blessing of Havinenu (and say), "Havinenu Hash-m Elokeinu ha'Mavdil Bein Kodesh l'Chol"!

This is left difficult.

Yoma 87b (Beraisa): On Motza'ei Yom Kipur, one prays seven Berachos that are like 18;

R. Chanina ben Gamliel says, he prays 18, so that he can say Havdalah in Chonen ha'Da'as.

Rishonim

Rambam (Hilchos Shabbos 29:28): When Yom Tov is in the middle of the week, on Motza'ei Yom Tov one says 'He who separates between Kodesh and profane, and between light and darkness, and between Yisrael and the Nochrim, and between Shabbos and the six days of work', just like one says on Motza'ei Shabbos. He need not bless on spices or on a Ner. Likewise, he need not bless on spices on Motza'ei Yom Kipur.

Tosfos (Beitzah 33b DH Ki): Why don't we smell spices on Yom Tov that is on Motza'ei Shabbos? Do not say that on a regular Motza'ei Shabbos we bless, because then the fire of Gehinom begins to burn and make a foul odor, but it does not burn on Yom Tov. Rav Yehudah holds that there is no concern lest one detach myrtle branches. If so, we should bless on Motza'ei Yom Tov! Do not say that we bless on Motza'ei Shabbos due to the extra Neshamah that we lost, and the spices revive us, and on Yom Tov there is an extra Neshamah, so we did not lose it when Shabbos ended. If so, we should bless on Motza'ei Yom Tov! Really, it is due to the loss of the extra Neshamah. There is no extra Neshamah on Yom Tov. When Motza'ei Shabbos is Yom Tov, since one has nice, good foods, this revives him without spices.

Tosfos (Pesachim 102b DH Rav): The Rashbam says that we bless on spices on Motza'ei Shabbos due to the loss of the extra Neshamah. This is not needed when Motza'ei Shabbos is Yom Tov, for also on Yom Tov there is an extra Neshamah. This is wrong. If so, we should bless on spices on Motza'ei Yom Tov!

Mordechai (Beitzah 693): Our Gemara refutes those who say that we do not bless on spices on Motza'ei Shabbos that is Yom Tov due to a decree lest one detach. We permit smelling, and we do not decree! This is the custom in Magentzia and Kolonya. R. Gershom and R. Meshulam explain like this. Rashi says that we do not bless. The Ri explains that the pleasure and Simchah of Yom Tov return the lost Neshamah.

Tosfos (Yoma 88a DH Mipnei): The Ri of Orleans asked why one must pray 18 Berachos on Motza'ei Yom Kipur. He should be able to include Havdalah in the middle Berachah of Havinenu, like it says in Berachos! The Ri answered that since the entire year one may pray Havinenu, we should not change, since he can include Havdalah in Havinenu. Here, he (R. Chanina) holds that the entire year one must pray 18 Berachos. Therefore, one should not change for a minor reason. Therefore, on Motza'ei Shabbos he prays 18, in order to say Havdalah in Chonen ha'Da'as.

Bach (OC 491:1 DH b'Motza'ei): Rav Nachman taught that one may not pray Havinenu on Motza'ei Shabbos or Motza'ei Yom Tov, for he must say Havdalah in Chonen ha'Da'as. Even though we challenged Rav Nachman and left the question difficult, he is not rejected, for the Gemara did not say 'Teyuvta' (refutation). Rav Hai Gaon, the Rif, Rambam, Rosh and Tur say so. This is unlike the Rashba, Ritz Gei'us and R. Yeshayah, who permit Havinenu on Motza'ei Shabbos and Motza'ei Yom Tov. It seems that Tosfos' text of Shmuel mentioned only Motza'ei Shabbos. They hold that Shmuel admits that one must prays 18 Berachos on Motza'ei Yom Tov in order to say Havdalah in Chonen ha'Da'as.

Note: Seemingly, the Bach should say that even Mar Zutra (who challenged Shmuel) admits that one must pray 18 Berachos on Motza'ei Yom Tov. Perhaps the Bach explains that Tosfos holds that Shmuel retracted due to Mar Zutra's question, but only regarding Motza'ei Shabbos.

Poskim

Shulchan Aruch (OC 491:1): On Motza'ei Yom Tov, whether it is Chol or Chol ha'Mo'ed, one says Havdalah like on Motza'ei Shabbos.

Beis Yosef (DH b'Motza'ei): On Motza'ei Yom Tov we pray like on Motza'ei Shabbos. I.e. we say Atah Chonantanu. Yoma 87b says that one must say the full Shemoneh Esre, for one must say Havdalah in Chonen ha'Da'as.

Shulchan Aruch (ibid.): However, he does not bless on light or spices.

Beis Yosef (DH u'Mah she'Chasav Ela): Since on Yom Tov one may light a Ner, there is no reason to bless on it in Havdalah!

Gra (2): The Amora'im argue only about Motza'ei Yom Kipur, but no one says to bless on a Ner on any other Motza'ei Yom Tov.

Beis Yosef (DH u'Mah she'Chasav v'Lo): The Rambam says that on Motza'ei Yom Tov, one need not bless on spices. Pesachim 102b connotes like this. Rav taught that when Yom Tov is on Motza'ei Shabbos, one blesses on the wine, Kidush, light and Havdalah. He did not mention spices. Tosfos concluded that there is no extra Neshamah on Yom Tov. The nice foods suffice in place of spices. The custom is like Tosfos, unlike the Mordechai.

Bach (DH u'Mah): The Mordechai mentioned a custom to bless on spices on Motza'ei Shabbos that is Yom Tov. The Gemara did not mention spices in the order of Havdalah because we never bless on a Ner without blessing also on spices. The custom is that we do not bless on spices.

Note: If we bless also on spices, the Gemara should have included this, to teach which comes first! Perhaps obviously the spices are before the Ner, like on a regular Motza'ei Shabbos.

Taz (1): There is no extra Neshamah on Yom Tov. The nice foods of Yom Tov revive a person like spices do.

Magen Avraham (1): The Simchah of Yom Tov revives the soul.

Sha'arei Teshuvah (1): Beis Yehudah (2:28) says that if one forgot to say Havdalah on Motza'ei Yom Tov, he can say it the entire week. Birkei Yosef (1) says that his proof is invalid.

Yabi'a Omer (7 OC 47:1): Beis Yehudah learns from the Shach (CM 43:48), who says that if a document says that it is to be paid after the festival, this is 15 days, which is the time for Olei Regalim to return to the end of Eretz Yisrael. However, Havdalah is only until Shabbos. Kisei Eliyahu says that one who forgot may say Havdalah for three days after Yom Tov, just like after Shabbos. This is like the Ritva (Gitin 77a), who says that 'after the festival' is three days, just like 'after Shabbos.' However, the Ramah says that it is half the time until the next festival, just like regarding Shabbos. Binyan Shlomo says that it should be until the next festival, just it is improper after the next Shabbos. Mekor Chayim says that there is Tashlumim only on the next day. Isru Chag is connected (tied) to the festival - "Isru Chag ba'Avosim..." Shabbos is different, for its Kedushah extends three days after it, like the Ari Zal brings from the Zohar.

Yabi'a Omer (ibid., 6): Beis Efrayim says that if one was an Onen or Anus, he may say Havdalah the next day. After that there is no Tashlumim. This is like the Rif says about Motza'ei Shabbos; the entire day is considered Motza'ei Shabbos. R. Akiva Eiger and many others agree. The Ben Ish Chai, Chesed l'Avraham and some others forbid after the night of Motza'ei Yom Tov. Had they seen the Rishonim (Ramah, R. Avraham bar Yitzchak and Kolbo) who permit the entire next day, they would have agreed. A Ben Yisrael in Chutz la'Aretz can try to hear Havdalah on Motza'ei Yom Tov Sheni from a Ben Chutz la'Aretz.

Radvaz (2:620): There is an extra Neshamah only on Shabbos, for it says "Shavas va'Yinafash." We do not bless on spices on Motza'ei Shabbos that is Yom Tov, for it would seem that he is in agony, and it says "v'Samachta b'Chagecha." Simchas Yom Tov removes agony.

Kaf ha'Chayim (297:1): The Bach says that we bless on spices only when two reasons apply: the extra Neshamah was lost, and the fire of Gehinom begins. Therefore, we do not bless on Motza'ei Shabbos that is Yom Tov (for there is no fire of Gehinom), and not on Motza'ei Yom Tov (no Neshamah was lost).

Kaf ha'Chayim (6): Machazik Berachah says that when we say Havdalah on Motza'ei Yom Tov, we do not mention Eliyahu.