After miraculously passing through the sea, and after coming to understand that their pursuers did did not fare likewise, the Jewish people are led by Moshe in prophetic song.
Towards the end of the song, when the people sing of the reactions of various nations to the great event, the Jewish nation sings of how the “the chieftains of Edom were confounded” and of how “trembling seized the powerful men of Moab.”
Having nothing to fear on account of Israel–those nations were not Cana’anites–Edom and Moab simply burned up at the glory and grandeur of the Jewish people. (Rashi)
Commentaries–speaking to why Edom and Moab, specifically, took unique umbrage at what had happened to the Jewish people–explain that the nations of Edom and Moab, stemming as they did from the family of Avraham, believed that they too had a claim to special honor. (Edom is Eisov, Jacob’s brother; and Moab is the son born of Lot’s (Avrahams nephew) incestuous relations with his eldest daughter.)
However, this raises a question as to why, despite equally illustrious lineage, the nations of Yishmael and Amon were not similarly bothered? (Yishmael was Isaac’s brother; Amon was Moab’s.)
The Lubavitcher Rebbe resolves the question thusly:
As for Yishmael:
When Genesis tells how “Isaac and Yishmael […] buried [Avraham] in the Makpelah cave”, the verse mentions the younger Isaac before the elder Yishmael. Commenting on scripture’s preferencing of Isaac, the Talmud explains that Torah wished to convey that although Yishmael had spent his whole life in contention with Isaac, by this point he had repented, and had himself preferenced Isaac in all of the procedures.
Thus, the nation of Yishmael, because of their patriarch, accepted Israel’s superiority.
And as for Amon:
These are the circumstances surrounding the birth of Moab and Amon:
Having narrowly escaped the destruction of Sodom, Lot and his two (remaining) daughters take shelter in a cave. Not knowing the death toll on humanity, Lot’s daughters begin to fear that they will never find men with whom to propagate. So following the plan of the eldest daughter–the future mother of Moab, and less modest of the two sisters–the young girls get their father drunk and lie with him. Moab and Amon are the fruit of that act.
Interestingly, the elder daughter feels no shame in the event. And when the time comes to name her son, she (proudly?) chooses “Moab”, a compound word made up of the Hebrew
for ‘from’ (mo) and ‘father’ (ab). The younger daughter, on the other hand, is far more discreet– a fact already alluded to in the wording of the verses–and when the time comes for her to name her son, she chooses “Amon”, a name derived from only part of the Hebrew phrase translating into ‘a member of my nation’ in English.
So while Moab, because of their shameless matriarch, proudly carried the banner of membership in Avraham’s family, Amon, because of their discreet matriarch, never made an explicit issue of that relationship, and therefore never internalized a claim to honor because of it.