What are the connotations of "Al-Pi Hashem"?
Rashi: With reference (not only to "Vaya'al Aharon", but also 1 ) to "Vayamos Sham" 2 - it teaches us that Aharon died through a kiss [from Hashem]. 3
Bava Basra, 17a: With reference (not only to "Vaya'al Aharon", but also) to "Vayamos Sham" - it teaches us that neither the Mal'ach ha'Ma'ves (during his death) nor the worms (after his death) had jurisdiction over Aharon.
Why does the Torah refer both here - regarding Av and in Devarim, 1:2 - regarding Sh'vat, as the fortieth year?
Rosh Hashanah, 2b: Because in the year that they left Egypt they counted the months from Nisan.
Yerushalmi Refer to Beha'aloscha, 9:1:3:1 and Torah Temimah, note 1.
Why does the Torah tell us the date of Aharon's death, and not that of Moshe?
Elef ha'Magen: Refer to 33:8:151:1. Because Aharon died in the month of Av, the month in which Yisrael cried for no reason and it was decreed that they would cry for future generations. Aharon's death hinted to a cessation of the Avodah (The Churbanos of both Batei-Mikdash occurred on Tish'ah be'Av). On the other hand, the Torah does not tell us the day that Moshe died because Torah, which is eternal, will never cease.
Oznayim la'Torah: Based on the Midrash Eichah - that following the episode of the Spies, every year on Tish'ah be'Av night, the men of that generation would dig a grave and lie in it, and in the morning they would discover that fifteen thousand plus had died, and that on Tish'ah be'Av of the fortieth year, nobody had died, the Torah is teaching us here that the decree came to an end one year prematurely due to the death of Aharon and the fact that the people wept at his death. 1
See Oznayim la'Torah, who elaborates and who comments that this also explains why the Torah mentions the date of his death here and not in Chukas, when the Torah records his death - because he died nine days before Tish'ah be'Av and the fact that the decree would be lifted was as yet unknown.
QUESTIONS ON RASHI
Rashi writes that Aharon died through a kiss. What are ramifications of this?
Moshav Zekenim #1: He put his lips together, and died - without any pain.
Moshav Zekenim #2: It was amidst clinging to Hashem and love of Him. Hashem placed His Shechinah on him, and he prophesied and died. "Kiss" hints at the way lovers kiss when they depart.
