When Tisha B’Ab falls on Shabbat and is thus observed on Mosa’eh Shabbat and Sunday, Habdala is not recited on Mosa’eh Shabbat. Only the Beracha over the candle – "Boreh Me’oreh Ha’esh" – is recited on Mosa’eh Shabbat, because this Beracha is specific to Mosa’eh Shabbat, and is what allows us to benefit from new light after Shabbat. We do not recite the Beracha over the Besamim (spices), because smelling the Besamim is a form of enjoyment, and we are to minimize our enjoyment on Tisha B’Ab.
On Sunday night, at the conclusion of Tisha B’Ab, we recite a brief Habdala, which consists of only the Beracha over wine and the Beracha of "Ha’mabdil." We do not recite the Beracha over the Besamim, because the smelling of Besamim was instituted as "compensation" for the loss of the Neshama Yetera ("extra soul") which we receive when Shabbat begins and then leaves after Shabbat. This Beracha is only relevant on Mosa’eh Shabbat, when the Neshama Yetera leaves, and thus when Tisha B’Ab begins on Mosa’eh Shabbat, such that we cannot recite the Beracha of Besamim, there is no reason to then recite it after Tisha B’Ab, on Sunday night. (This is similar to the case when Yom Tob begins on Mosa’eh Shabbat, and we combine Habdala with the Kiddush for Yom Tob – in a sequence known as "Yaknehaz" – and we omit the Beracha of Besamim. Some say this is because we have a Neshama Yetera on Yom Tob, too, and others explain that festive Yom Tob meal suffices to fill the "vacuum" created by the departure of the extra soul. And since Besamim is not necessary on Mosa’eh Shabbat, we do not recite the Beracha the next night, either. The same applies when Tisha B’Ab begins on Mosa’eh Shabbat.)
Additionally, when we recite Habdala on Sunday night after the conclusion of Tisha B’Ab, we omit the introductory verses. These verses, which are expressions of our hopes for good fortune, are normally recited as part of Habdala in order to arouse joy and festivity as we begin the new week, and so, like the Besamim, once they are not recited on Mosa’eh Shabbat, there is no purpose in reciting them on Sunday night. And so we begin Habdala after Tisha B’Ab with the words, "Kos Yeshu’ot Esa," and we then lift the cup, recite "Sabri Maranan," followed by the Beracha over the wine and the Beracha of "Ha’mabdil."
When Tisha B’Ab begins on Mosa’eh Shabbat, those who are permitted to eat on Tisha B’Ab must recite Habdala before eating. (When Tisha B’Ab falls on Shabbat and is thus delayed until Sunday, there is greater room for leniency when it comes to ill patients, and pregnant and nursing women, as in some cases, those who normally would be required to fast on Tisha B’Ab are allowed to eat on Tisha B’Ab which is observed on the 10th of Ab.) Hacham Ovadia Yosef ruled that they should recite Habdala immediately on Mosa’eh Shabbat, and not wait until they need to eat. They omit the introductory verses and Besamim, and so they recite only the Beracha over wine, the Beracha over the candle, and "Ha’mabdil."
Summary: When Tisha B’Ab is observed on Mosa’eh Shabbat and Sunday, the blessing over the candle is recited on Mosa’eh Shabbat, and on Sunday night, at the conclusion of Tisha B’Ab, we recite only the Beracha over the wine and "Ha’mabdil." The rest of Habdala – the introductory verses and Besamim – is not recited at all, neither on Mosa’eh Shabbat nor on Sunday night. An ill patient, or a pregnant or nursing woman, who is allowed to eat on Tisha B’Ab, recites Habdala on Mosa’eh Shabbat, omitting the introductory verses and Besamim.
