R' Akiva Eger (teshuva #24) raises the issue can someone not fasting receive an aliyah on Yom Kippur at mincha - is the reading a kedushat hayom Yom Kippur reading or a fast day reading? The Marcheshes siman 14 says that issue is the root of the machlokes between the Mechabar and Remah (622:2) if the bracha of the haftorah ends מגן דוד or continues with על התורה ועל העבודה like a holiday. the Rema that ends מגן דוד holds that the reading is that of a fast day so we don't mention the day but the Mechaber holds we do since it is the reading of a holiday. This understanding of the Rema is explicit in the Gr"a who says that the reading and haftorah are due to the day being a fast day. However, the Levush, MG"A and Alter Rebbe explain the Rema differently. They explain that the avodah of the day is already complete before mincha therefore we don't say על התורה ועל העבודה.
A parallel inquiry applies to the ne'ilah prayer, which conceptually exists for any public fast day Is the נעילה on Yom Kippur a YK prayer or that of a taanis? R' Zvi Ryzman raises the question regarding a kohan that is not fasting can he do נשיאת כפים by ne'ilah? If ne'ilah is fundamentally a fast-day prayer, a non-fasting kohan would be disqualified; however, if it is defined by the kedushat hayom of Yom Kippur, he may be permitted to recite the blessing.
In Kol Brisk he explains why in the ne'ilah prayer we only say the short viduy of אשמנו and not the longer על חטא recitation. He says that the על חטא prayers acknowledge wrongdoings but don't describe the person himself as a bad person, just that he has strayed in various aspects. אשמנו is a description of the person as affected by sin. On a fast day we reflect on our sad state as sinners. We do not say words of request for forgiveness. On YK however the viduy is the long one. we insert requests for forgiveness for our focus is לפני ה תטהרו how we remain connected to Hashem and just have to mend our ways. When it comes to נעילה we are davening a prayer of a taanis and hence we omit the long viduy.
On the other hand, the Chiunuch (#433) views the ne'ilah prayer as a YM prayer- וְעוֹד תִּקְּנוּ תְּפִלָּה חֲמִישִׁית בְּיוֹם הַכִּפּוּרִים לְבַד, לְרֹב קְדֻשַּׁת הַיּוֹם, וּבַעֲבוּר הֱיוֹתוֹ יוֹם סְלִיחָה וְכַפָּרָה לַכֹּל, וְהִיא הַנִּקְרֵאת תְּפִלַּת נְעִילָה. For more discussion about this topic see תפילת נעילה ביום הכיפורים.
Ultimately, Yom Kippur shares the element of a standard fast day and is viewd as an עת צרה. Yet through its unique sanctity, we are granted the capacity for purification before Hashem, לפני ה תטהרו, transforming צרה, into צהר, a window of illumination, as an opportunity to begin anew with a new light unto our lives.