Psalm 49:2-3 reads something like:
Hear inhabitants of the world, (and) also sons of Adam, and also sons of Ish...
Why is there a distinction made here between inhabitants of the world, sons of Adam and sons of Ish?
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It is because each of these three groupings serve a different message.
This first classification is a general labelling of the failing of mankind and their pursuit of material possessions (See Rashi).
Metzudas Dovid writes expressly:
כל העמים. כי כולם צריכים לדעת המוסר הזה:
"All you people" - because everybody needs this lesson.
Malbim adds that this is an all-encompassing rebuke that includes both people and nations that make up a society, as well as individuals, who all err in this area.
Rashi sees it as a reference to those descended from Avraham, namely Yishmael and the sons of Keturah. The reason why they have the title בְּנֵ֣י אָ֭דָם is because Avraham is referenced as "the greatest man (האדם) among the giants" (Yehoshua 14:15)
Alternatively, Shadal (accessible here) writes that it is to be understood in a more literal sense i.e. that of mankind which comes from Adam:
אותם שאין להם מעלה אחרת אלא שהם מהמין האנושי בני אדם וחוה
Those who have no other virtue but are of the human race, i.e. the sons of Adam and Chava
Alternatively, the Targum notes that this is a reference to the sons of Yaakov who are "כַּחֲדָא זַכָּאָה וְחַיָבָא" - "Like one - those with merits and those who are (lit.) obligated". The Midrash understands it to mean that those who are both rich and impoverished in Torah study are together as one, as both have the capacity to descend to Gehinnom. The Midrash notes how both Doeg and Achisophel were considered rich in Torah as they were both heads of the Sanhedrin however they did not do enough with this 'wealth' and were accordingly destined to fail. Similarly, those who are 'poor' means those who have the ability to learn but choose not to. In both instances these people are accountable for their actions and will descend together.
Shadal (accessible here) writes that it is a reference to those who have a greater societal standing:
אותם שאביהם או אחד מזקניהם היה איש מיוחד ונכבד הנבדל ונודע מצד מעלתו, כמו הלא איש אתה ומי כמוך בישראל (שמואל א כ״ו:ט״ו). והנה תחלה הזכיר החלוק שבין בני אדם מצד אבותיהם, ואחר כך הזכיר ההפרש שביניהם מצד הממון, ואמר יחד עשיר ואביון
Those whose father or one of their elders was a special and honourable man who was distinguished and known due to his virtues. Like, "Are you not the man, and who is like you Israel" (Shmuel I 26:15). So here it begins by mentioning a distinction between 'sons of man' from the side of their fathers (i.e. their yichus), and afterwards differentiates from the perspective of their wealth, thus it says "rich and poor alike".
Building on what I quoted from Shadal, a general rule begins to appear. It is worth noting that it says in Tehillim 4:3
בְּנֵ֥י אִ֡ישׁ עַד־מֶ֬ה כְבוֹדִ֣י לִ֭כְלִמָּה תֶּאֱהָב֣וּן רִ֑יק תְּבַקְשׁ֖וּ כָזָ֣ב סֶֽלָה
You men, how long will my glory be mocked, will you love illusions, have recourse to frauds? Selah. (Sefaria translation)
Radak writes there an interesting definition that is pertinent to your question.
בני איש אמר כנגד גדולי ישראל שהיו עם אבשלום. וכן גם בני אדם גם בני איש (תהלים מט ג). בני אדם ההמון, בני איש הגדולים. וכן הלוא איש אתה ומי כמוך בישראל (שמואל א כו טו); גדעון בן יואש איש ישראל (שופטים ז יד).
O ye sons of men (בני איש):He addresses the great ones of Israel who were with Absalom. So "both low (בני אדם) and high (אישׁ בני)" (Ps. 49:3): בני אדם (are) the multitude, אישׁ בני the great ones. So also, "Art not thou a great man (אישׁ) and who is like to thee in Israel?" (1 Sam. 26:15); "Gideon the son of Joash, the great man (אישׁ) 0f Israel" (Judges 7:14) (Sefaria translation).
So it would seem from this that בני אדם is generally a reference to the many whilst בני איש is attributed to those of greater stature. This idea is also supported by the Metzudas Zion on the pasuk you bring.