The historical interpretation
Aaron did not die.
Aaron did not exist. He is as mythological as Moses, Exodus or everything chronologically before that.
The literary interpretation of two different texts
Concurring with the historical interpretation:
The historicity of Moses has been much discussed by OT scholars. Generally, it is accepted that there are three sources in the narrative known by the symbols J, E, and P, […]; but there is no confirmation of Moses' existence from archaeology or other ancient Near Eastern documents though the name, as also Aaron, is of Egyptian origin. We are left only with the view held about the internal evidence of the Pentateuch itself, where there are inconsistencies and doubts.
–– Moses Source: A Dictionary of the Bible, oxford biblical studies online
… and this is one of those inconsistencies.
According to the biblical traditions, the Israelites journeyed from Kadesh to Mount Hor (whose location is unknown), where Aaron died, and they then defeated the king of Arad at Hormah (Num. 20: 22–21: 3). Thereafter, it is difficult to be clear what route is envisaged by those who recounted the traditions. A summary account of the route (Num. 33: 41–9) describes them as heading relatively directly for Moab, via Zalmonah, Punon, and Oboth. However, Numbers 21: 4 suggests a detour via the Re(e)d Sea, that is, probably Ezion-geber and the Gulf of Aqaba (see Deut. 2: 8), and an avoidance of the land of Edom. They subsequently encamped in the Wadi Zered, and then crossed the River Arnon and continued on to the vicinity of Mount Pisgah (Num. 21: 10–20).
–– Adrian Curtis: "Oxford Bible Atlas", Oxford University Press: Oxford, New York, 42007.
In stark contrast to
Moserah – Place where Aaron died (Deut 10:6). Evidently near or identical to Mt. Hor (Num 20 passim).
Moseroth – Israelite campsite between Hashmonah and Bene Jaakan (Num 33:30, 31)–Unknown; in Sinai or S Negev. Some suggest identical to Moserah.
Hor, Mount (Lebanon) – Mountain on N border of land of Canaan (Num 34:7, 8).–Uncertain; possibly Ras Shaqqa (213412), 30 mi. NNE of Beirut on Mediterranean coast.
Hor, Mount (Negev) – Israelite campsite during wilderness wanderings. There Aaron died (Num 20 passim, etc.). Mentioned 10 times in OT.–Uncertain; possibly Imaret el-Khureisheh (104017), 40 mi. SW of Beer Sheba.
–– Carl G. Rasmussen: "Atlas Of The Bible", Zondervan: Grand Rapids, 2010.
These are indeed very different places. The simplest solution is of course that we read two very different texts and that those are just not in agreement. If the places are identical and important, why change the names? If the names are just invented anyway, why not just use those that fit your intentions?
While both places cannot be identified with certainty, the differences displayed in the stations made make this 'solution' the preferred one.
Two different sources, two different aims for the texts. With Dtn 10 being a very late addition to the canon (Ulrich Dahmen: "Leviten und Priester im Deuteronomium: Literarkritische und redaktionsgeschichtliche Studien", 1996.)
If you want a less scholarly solution but more doctrinally contradiction-'free' one then you might want to go with a rabbinical explanation:
Death of Aaron.
In fulfilment of the promise of peaceful life, symbolized by the pouring of oil upon his head (Lev. R. x., Midr. Teh. cxxxiii. 1), Aaron's death, as described in the Haggadah, was of a wonderful tranquillity. Accompanied by Moses, his brother, and by Eleazar, his son, Aaron went to the summit of Mount Hor, where the rock suddenly opened before him and a beautiful cave lit by a lamp presented itself to his view. "Take off thy priestly raiment and place it upon thy son Eleazar!" said Moses; "and then follow me." Aaron did as commanded; and they entered the cave, where was prepared a bed around which angels stood. "Go lie down upon thy bed, my brother," Moses continued; and Aaron obeyed without a murmur. Then his soul departed as if by a kiss from God. The cave closed behind Moses as he left; and he went down the hill with Eleazar, with garments rent, and crying: "Alas, Aaron, my brother! thou, the pillar of supplication of Israel!" When the Israelites cried in bewilderment, "Where is Aaron?" angels were seen carrying Aaron's bier through the air. A voice was then heard saying: "The law of truth was in his mouth, and iniquity was not found on his lips: he walked with me in righteousness, and brought many back from sin" (Mal. ii. 6, 7). He died, according to Seder 'Olam R. ix., R. H. 2, 3a, and Josephus, "Ant." iv. 4, § 7, on the first of Ab. Josephus says also that "he died while the multitude looked upon him." The pillar of cloud which proceeded in front of Israel's camp disappeared at Aaron's death (see Seder 'Olam, ix. and R. H. 2b-3a). The seeming contradiction between Num. xx. 22 et seq. and Deut. x. 6 is solved by the rabbis in the following manner: Aaron's death on Mount Hor was marked by the defeat of the people in a war with the king of Arad, in consequence of which the Israelites fled, marching seven stations backward to Mosera, where they performed the rites of mourning for Aaron; wherefore it is said: "There [at Mosera] died Aaron." See Mek., Beshallaḥ, Wayassa', i.; Tan., Huḳḳat, 18; Yer. Soṭah, i. 17c, and Targ. Yer. Num. and Deut. on the abovementioned passages.
The rabbis also dwell with special laudation on the brotherly sentiment which united Aaron and Moses. When the latter was appointed ruler and Aaron high priest, neither betrayed any jealousy; instead they rejoiced in one another's greatness. When Moses at first declined to go to Pharaoh, saying: "O my Lord, send, I pray thee, by the hand of him whom thou wilt send" (Ex. iv. 13), he was unwilling to deprive Aaron, his brother, of the high position the latter had held for so many years; but the Lord reassured him, saying: "Behold, when he seeth thee, he will be glad in his heart" (Ex. iv. 14). Indeed, Aaron was to find his reward, says Simon ben Yoḥai; for that heart which had leaped with joy over his younger brother's rise to glory greater than his was decorated with the Urim and Thummim, which were to "be upon Aaron's heart when he goeth in before the Lord" (Cant. R. i. 10). Moses and Aaron met in gladness of heart, kissing each other as true brothers (Ex. iv. 27; compare Song of Songs, viii. 1), and of them it is written: "Behold how good and how pleasant [it is] for brethren to dwell together in unity!" (Ps. cxxxiii. 1). Of them it is said (Ps. lxxxv. 10): "Mercy and truth are met together; righteousness and peace have kissed [each other]"; for Moses stood for righteousness, according to Deut. xxxiii. 21, and Aaron for peace, according to Mal. ii. 6. Again, mercy was personified in Aaron, according to Deut. xxxiii. 8, and truth in Moses, according to Num. xii. 7 (Tan., Shemot, ed. Buber, 24-26).
When Moses poured the oil of anointment upon the head of Aaron, Aaron modestly shrank back and said: "Who knows whether I have not cast some blemish upon this sacred oil so as to forfeit this high office." Then the Holy Spirit spake the words: "Behold the precious ointment upon the head, that ran down upon the beard of Aaron, that even went down to the skirts of his garment, is as pure as the dew of Hermon" (Ps. cxxxiii. 2, 3, Heb.; Sifra, Shemini, Milluim; Tan., Korah, ed. Buber, 14).
ESV: Num –– 28 And they set out from Terah and camped at Mithkah. 29 And they set out from Mithkah and camped at Hashmonah. 30 And they set out from Hashmonah and camped at Moseroth. 31 And they set out from Moseroth and camped at Bene-jaakan. (De 10:6) 32 And they set out from Bene-jaakan and camped at Hor-haggidgad. (De 10:7) 33 And they set out from Hor-haggidgad and camped at Jotbathah. (Nu 33:32) 34 And they set out from Jotbathah and camped at Abronah. 35 And they set out from Abronah and camped at Ezion-geber. (De 2:8; 1Ki 9:26; 1Ki 22:48; 2Ch 8:17) 36 And they set out from Ezion-geber and camped in the wilderness of Zin (that is, Kadesh). (Nu 20:1; Nu 27:14) 37 And they set out from Kadesh and camped at Mount Hor, on the edge of the land of Edom. (Nu 20:1; Nu 20:14; Nu 20:22; Nu 20:23; Nu 21:4; Nu 34:7) 38 And Aaron the priest went up Mount Hor at the command of the Lord and died there, in the fortieth year after the people of Israel had come out of the land of Egypt, on the first day of the fifth month. (Nu 20:25; De 10:6; De 32:50) 39 And Aaron was 123 years old when he died on Mount Hor. (Ex 7:7) 40 And the Canaanite, the king of Arad, who lived in the Negeb in the land of Canaan, heard of the coming of the people of Israel. (Nu 21:1) 41 And they set out from Mount Hor and camped at Zalmonah. 42 And they set out from Zalmonah and camped at Punon.
Compared to Dtn
ESV: (The people of Israel journeyed from Beeroth Bene-jaakan4 to Moserah. There Aaron died, and there he was buried. And his son Eleazar ministered as priest in his place. (Nu 20:28; Nu 33:30; Nu 33:38) 7 From there they journeyed to Gudgodah, and from Gudgodah to Jotbathah, a land with brooks of water.
For Num we see:
Moseroth –– Bene-jaakan –– Hor-haggidgad –– Jotbathah –– Abronah –– Ezion-geber –– Zin (that is, Kadesh) –– Mount Hor (Aaron dies)
For Dtn we see:
Beeroth Bene-jaakan –– Moserah (Aaron dies) –– Gudgodah –– Jotbathah
According to Numbers 33:31-32, the Israelites "departed from Moseroth and camped at Bene Jaakan, and then moved from Bene Jaakan and camped at Hor Hagidgad", but according to Deuteronomy 10:6, "the children of Israel journeyed from the wells of Bene Jaakan to Moserah" (or Mosera) after the second occasion when Moses had been given the Ten Commandments inscribed on two tablets of stone. The stage reported in the Book of Numbers represents two legs among 24 recorded in summary in Numbers 33:15-37 whereas at this point in the Deuteronomy narrative only four stations of the Exodus are mentioned (Bene Jaakan, Moserah, Gudgodah and Jotbathah).
The two different texts show two different stories. The two texts by themselves are absolutely irreconcilable on a factual level. One says death on Mount Hor, the other says death at Moserah. One says before Jotbathah, the other says after. One says from Moseroth to Bene-jaakan, the other from Beeroth Bene-jaakan to Moserah. In one version the group effectively goes back eight stations to let Aaron die? Or did he just die twice?
If you want to get creative again in reading both stories, you may want to reconsider that neither of the two books is a history book. They both retell a foundational myth and do that by different means.
As Rabbi Moshe Ben-Chaim explains:
Aaron did in fact die on Mount Hor […]. How then can Deuteronomy state that he died in Mosayra, a campsite the Jews left eight journeys earlier? Numbers 33:31-37 chronologically records eight campsites traveled by the Jews. It commences with Mosayra, and concludes eight locations later with Mount Hor. 33:38 openly states that Aaron died on Mount Hor, a completely different location than Mosayra.
The Torah is not a history book. […] What is Moses’ lesson here? Moses says, “Aaron died there at Mosayra” when we know in fact that he died in Mount Hor. But Moses did so since he is their Rebbe and leader: the greatest teacher of human perfection. He describes Aaron’s death “as if it was in Mosayra”, to indicate the “cause” of why the Jews found themselves back at Mosayra.
Moses subtly taught the Jews that he attributed their national reversion to Egypt (by way of Mosayra) to be in connection with Aaron’s death, and the departing of God’s protective clouds at that time. As Rashi teaches, the Jews were then fearful of warring with the King of Ard. They headed back towards Egypt. This was a rebellion, and it required a rebuke. But instead of openly stating this rebuke as Moses did when describing the breaking of the Tablets, here, Moses used a subtle hint. Why? Perhaps this rebuke required more understanding by the Jews, as their sin was not as overt as prostrating to a Golden Calf. That sin could be addressed openly, since no one could deny his or her corruption. But on the surface, “traveling backwards” does not appear as sinful. In order to engage the mind of the Jews, Moses created an apparent contradiction in Aaron’s place of death, which would awaken the Jews to ponder that location of Mosayra, and hopefully, awaken them to consider ‘why’ they arrived there. They might now consider that earlier event, and their rebellious nature. Joining Aaron’s death with the rebuke of the broken Tablets, Moses helped the Jews associate Aaron’s death and their return to Mosayra, with the sin of the Tablets and the Golden Calf. They might then view their return to Mosayra in the same sinful light, and unveil for themselves their national error.
Their desire to return to Egypt – why they were back at Mosayra – should also make the Jews realize their attachment to Egypt. But if Moses openly rebuked the Jews, and did not allow them to consider apparently contradictory burial sites of Aaron, their minds would be less engaged, and they would not ponder that return to Mosayra, with all of its ramifications. They would not reflect as much, and they would neglect to grasp their attachment to Egypt. Rashi concluded with these words:
“…it was as difficult to Him when they said, ‘set us a leader to return to Egypt’, as was the day they forged the Golden Calf.”
Rashi teaches us that the Jews’ return to Mosayra – a mere stop along the way back to Egypt – carried a sin equal to the Golden Calf. Just as the Golden Calf expressed idolatrous tendencies, surely their return to the origin of calf worship – Egypt – expressed their sustained, idolatrous attachment.
The Torah is not a history book, so its text must be studied, together with the counsel of the Sages’ words, and not simply read.
This explanation is indeed quite nicely fitting with a compositional date of Deuteronomy after king Josiah's reforms to introduce monotheistic policy into the realm which demands in the shema Yisrael the exclusivity of JHWH worship (monolatry or mono- or henotheism, or "cult purity", and the newly invented basic commandment of Deuteronomy now requiring the centralization of the cult at the Jerusalem temple or "cult unity".
–– Nadav Na'Aman: "The "Discovered Book" and the Legitimation of Josiah's Reform", Journal of Biblical Literature, Vol. 130, No. 1, 2011, pp. 47-62.
Different books, different authors, different redactors, different intents, different statements, different stories.