Korach - For the sake of heaven

(א) וַיִּקַּ֣ח קֹ֔רַח בֶּן־יִצְהָ֥ר בֶּן־קְהָ֖ת בֶּן־לֵוִ֑י וְדָתָ֨ן וַאֲבִירָ֜ם בְּנֵ֧י אֱלִיאָ֛ב וְא֥וֹן בֶּן־פֶּ֖לֶת בְּנֵ֥י רְאוּבֵֽן׃ (ב) וַיָּקֻ֙מוּ֙ לִפְנֵ֣י מֹשֶׁ֔ה וַאֲנָשִׁ֥ים מִבְּנֵֽי־יִשְׂרָאֵ֖ל חֲמִשִּׁ֣ים וּמָאתָ֑יִם נְשִׂיאֵ֥י עֵדָ֛ה קְרִאֵ֥י מוֹעֵ֖ד אַנְשֵׁי־שֵֽׁם׃ (ג) וַיִּֽקָּהֲל֞וּ עַל־מֹשֶׁ֣ה וְעַֽל־אַהֲרֹ֗ן וַיֹּאמְר֣וּ אֲלֵהֶם֮ רַב־לָכֶם֒ כִּ֤י כָל־הָֽעֵדָה֙ כֻּלָּ֣ם קְדֹשִׁ֔ים וּבְתוֹכָ֖ם יְהוָ֑ה וּמַדּ֥וּעַ תִּֽתְנַשְּׂא֖וּ עַל־קְהַ֥ל יְהוָֽה׃ (ד) וַיִּשְׁמַ֣ע מֹשֶׁ֔ה וַיִּפֹּ֖ל עַל־פָּנָֽיו׃

Numbers 16:1-4

(1) Now Korah, son of Izhar son of Kohath son of Levi, betook himself, along with Dathan and Abiram sons of Eliab, and On son of Peleth—descendants of Reuben— (2) to rise up against Moses, together with two hundred and fifty Israelites, chieftains of the community, chosen in the assembly, men of repute. (3) They combined against Moses and Aaron and said to them, “You have gone too far! For all the community are holy, all of them, and the LORD is in their midst. Why then do you raise yourselves above the LORD’s congregation?” (4) When Moses heard this, he fell on his face.

What was the point of Korach's argument? Why did he disagree with Moses?

משנה אבות ה׳:י״ז

(יז) כָּל מַחֲלֹקֶת שֶׁהִיא לְשֵׁם שָׁמַיִם, סוֹפָהּ לְהִתְקַיֵּם. וְשֶׁאֵינָהּ לְשֵׁם שָׁמַיִם, אֵין סוֹפָהּ לְהִתְקַיֵּם. אֵיזוֹ הִיא מַחֲלֹקֶת שֶׁהִיא לְשֵׁם שָׁמַיִם, זוֹ מַחֲלֹקֶת הִלֵּל וְשַׁמַּאי. וְשֶׁאֵינָהּ לְשֵׁם שָׁמַיִם, זוֹ מַחֲלֹקֶת קֹרַח וְכָל עֲדָתוֹ:

Pirkei Avot 5:17

(17) Every dispute that is for the sake of Heaven, will in the end endure; But one that is not for the sake of Heaven, will not endure. Which is the controversy that is for the sake of Heaven? Such was the controversy of Hillel and Shammai. And which is the controversy that is not for the sake of Heaven? Such was the controversy of Korah and all his congregation.

What does it mean to be a controversy for the sake of Heaven?

Note that for the Rabbis of the Talmud, a Makhloket l'shem shamayim, an argument for the sake of heaven, is not one in which anyone gives in, nor one where one refused to hear the other side. It is a disagreement in which people are polite and in which both sides are open to hearing the other side. In order to truly place others' words before our own, as Beit Hillel did, we must listen to them and be able to restate them before we respond to them. -- Johannah Sohn

What's the difference between the two examples from Pirkei Avot 5:17?
What can we learn from these examples? From Korach's argument?

ברטנורא על משנה אבות ה׳:י״ז:א׳

(א) כָּל מַחֲלֹקֶת שֶׁהִיא לְשֵׁם שָׁמַיִם סוֹפָהּ לְהִתְקַיֵּם. כְּלוֹמַר שֶׁאַנְשֵׁי הַמַּחֲלֹקֶת הַהִיא מִתְקַיְּמִים וְאֵינָם אוֹבְדִין, כְּמַחֲלֹקֶת הִלֵּל וְשַׁמַּאי שֶׁלֹּא אָבְדוּ לֹא תַּלְמִידֵי בֵּית שַׁמַּאי וְלֹא תַּלְמִידֵי בֵּית הִלֵּל. אֲבָל קֹרַח וַעֲדָתוֹ אָבְדוּ...וְהַמַּחֲלֹקֶת שֶׁהִיא לְשֵׁם שָׁמַיִם, הַתַּכְלִית וְהַסּוֹף הַמְבֻקָּשׁ מֵאוֹתָהּ מַחֲלֹקֶת לְהַשִּׂיג הָאֱמֶת, וְזֶה מִתְקַיֵּם, כְּמוֹ שֶׁאָמְרוּ מִתּוֹךְ הַוִּכּוּחַ יִתְבָּרֵר הָאֱמֶת, וּכְמוֹ שֶׁנִּתְבָּאֵר בְּמַחֲלֹקֶת הִלֵּל וְשַׁמַּאי שֶׁהֲלָכָה כְּבֵית הִלֵּל. וּמַחֲלֹקֶת שֶׁאֵינָהּ לְשֵׁם שָׁמַיִם, תַּכְלִית הַנִּרְצֶה בָּהּ הִיא בַּקָּשַׁת הַשְּׂרָרָה וְאַהֲבַת הַנִּצּוּחַ, וְזֶה הַסּוֹף אֵינוֹ מִתְקַיֵּם, כְּמוֹ שֶׁמָּצִינוּ בְּמַחֲלֹקֶת קֹרַח וַעֲדָתוֹ שֶׁתַּכְלִית וְסוֹף כַּוָּנָתָם הָיְתָה בַּקָּשַׁת הַכָּבוֹד וְהַשְּׂרָרָה וְהָיוּ לְהֶפֶךְ:

Bartenura on Pirkei Avot 5:17:1

Every argument that is for [the sake of] heaven's name, it is destined (literally, its end is) to endure: That is to say that the [parties to] the argument are destined to endure and not perish, as with the argument between Hillel and Shammai, [whereby] neither the students of the School of Hillel nor the students of the School of Shammai perished. But Korach and his congregation perished...The argument which is for the sake of Heaven, the purpose and aim that is sought from that argument is to arrive at the truth, and this endures; like that which they said, "From a dispute the truth will be clarified," and as it became elucidated from the argument between Hillel and Shammai - that the law was like the school of Hillel. And [with] argument which is not for the sake of Heaven, its desired purpose is to achieve power and the love of contention, and its end will not endure; as we found in the argument of Korach and his congregation - that their aim and ultimate intent was to achieve honor and power, and the opposite was [achieved].