oomis (and anyone else confused with this), the reason why we make a Borei Meore HaEsh on Motzei Shabbos is because Adam first “discovered” fire on Motzei Shabbos when the Ohr Haganuz disappeared at the end of the first Shabbos. So, as a praise and thanks to Hashem, we light a fire on Motzei Shabbos and make a Bracha.
Yom Kippur has a different reason. On all Yomim Tovim we are allowed to cook and use fire for “ochel nefesh”. On Yom Kippur all melacha is forbidden (like Shabbos), making it stricter then all other Yomim Tovim. Because we could not use fire on Yom Kippur, we light a candle and make Borei Meore HaEsh to show that fire is now permitted to be used.
This is only done on a “Ner ShShavas”, literally, light that rested, meaning, a candle that was lit before Yom Kippur. This is to show that we had a lit candle avaliable and we did not use it because on Yom Kippur we are forbidden from all melacha, even ochel nefesh. If one has no Ner SheShavas he does not make a Borei Meore HaEsh. (There is a mitzvah to find one and make the Bracha.)
However, when Yom Kippur is also Shabbos, we have both reason to make the Bracha, Motzei Shabbos and Motzei Yom Kippur. In this case one need not use a Ner SheShavas but can light a new fire and make the Bracha. It is preferable to use a Ner SheShavas so you get both reasons.
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Besamim is not used on a regular Motzei Yom Kippur but on Motzei Shabbos Yom Kippur it is used. Hinei Kel (until Borei Pri HaGofen) is omitted on a regular Motzei Yom Kippur but on Motzei Shabbos Yom Kippur it is recited.
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I have found it hard to light a Havdala candle from the bottom of the Yartzeit (24 hour) candle because the wax has basically run out and the flame is fighting to stay alive. When I try lighting the candle, the flame goes out.
I thought of a solution to this problem. Instead of using a regular 24 hour candle, use a 48 hour one stop this way the candle has plenty wax underneath it and is not struggling to stay alive. The only problem with this is, what to do with the rest of the candle? I saw that it is not good to extinguish a candle that was burning all Yom Kippur. I may be wrong. I have not tested this idea out yet.