Our Gemara on Amud Aleph mentions the phrase, כְּדֵי שְׁאֵילַת שָׁלוֹם תַּלְמִיד לָרַב the time necessary for a student to inquire after the welfare of his rabbi. The Rishonim (see Shitta Mekubetzes here, Tosafos here 20b “amar ley”, and Gemara Bava Kama 73b) delineate the phrase as, “Peace onto you, my master”.
The nature of this greeting is debated in halakha because of its tone of familiarity. That is, the idea of offering a greeting to the master as if he is a peer itself might be problematic, as well as it being addressed directly (“onto you”) instead of in third person, (“onto the master”). See Shulkhan Arukh 242:16 and Arukh Hashulkhan YD 242:39.
Though in regard to parents there are extensive requirements to show respect (ibid, 240), there is more room for warmth and familiarity. We can see this represented by comparing the dialogue of Yaakov and his father, versus the dialogue of Esav and his father.
Yaakov addresses his father as follows:
וַיֹּ֨אמֶר יַעֲקֹ֜ב אֶל־אָבִ֗יו אָנֹכִי֙ עֵשָׂ֣ו בְּכֹרֶ֔ךָ עָשִׂ֕יתִי כַּאֲשֶׁ֥ר דִּבַּ֖רְתָּ אֵלָ֑י קֽוּם־נָ֣א שְׁבָ֗ה וְאׇכְלָה֙ מִצֵּידִ֔י בַּעֲב֖וּר תְּבָרְכַ֥נִּי נַפְשֶֽׁךָ׃
Jacob said to his father, “I am Esau, your first-born; I have done as you told me. Please, come and recline to eat of my game, that you may give me your innermost blessing.” (Bereishis 27:19)
Esav addresses his father as follows:
וַיַּ֤עַשׂ גַּם־הוּא֙ מַטְעַמִּ֔ים וַיָּבֵ֖א לְאָבִ֑יו וַיֹּ֣אמֶר לְאָבִ֗יו יָקֻ֤ם אָבִי֙ וְיֹאכַל֙ מִצֵּ֣יד בְּנ֔וֹ בַּעֲבֻ֖ר תְּבָרְכַ֥נִּי נַפְשֶֽׁךָ׃
He too prepared a dish and brought it to his father. And he said to his father, “Let my father sit up and eat of his son’s game, so that you may give me your innermost blessing. (Ibid 31.)
Yaakov uses a familiar first person mode of speech, while Esau speaks in indirectly (“Let my father get up.”) We have a tradition that Esav excelled in honoring his parents, as Rabbi Shimon Ben Gamliel remarks (Bereishis Rabbah 65:16): “I have served my father my entire life but have not met 1/100 of the measure that Esau served.” We might argue therefore that Esau here too was extra careful in his formality.
However, that is not how Chasam Sofer (Toldos) sees it. He quotes his rebbe, Rav Nosson Adler, that Yaakov spoke with warmth toward his father, and Esau’s terminology was cold and detached. A similar idea is expressed by the Zohar (I:144a)
וַיַּעַשׂ גַּם הוּא מַטְעַמִּים וְגו’. (בראשית כ״ז:ל״א) יָקוּם אָבִי, דִּבּוּרֵיהּ הֲוָה בְּעַזּוּת, בִּתְקִיפוּ רוּחָא, מִלָּה דְּלֵית בָּהּ טַעֲמָא, יָקוּם אָבִי. תָּא חֲזֵי, מַה בֵּין יַעֲקֹב לְעֵשָׂו, יַעֲקֹב אָמַר בִּכְסִיפוּ דְּאֲבוֹי, בַּעֲנָוָה, מַה כְּתִיב, (בראשית כ״ז:י״ח) וַיָּבֹא אֶל אָבִיו וַיֹּאמֶר אָבִי. מַה בֵּין הַאי לְהַאי, אֶלָּא, דְּלָא בָעָא לְאִזְדַּעְזְעָא לֵיהּ, מַלִּיל בְּלָשׁוֹן תַּחֲנוּנִים, (בראשית כ״ז:י״ט) קוּם נָא שְׁבָה וְאָכְלָה מִצֵּידִי. וְעֵשָׂו אָמַר, יָקוּם אָבִי, כְּמַאן דְּלָא מַלִּיל עִמֵּיהּ.
And he also had made savory food…Let my father arise” (Beresheet 27:31). His speech was impertinent, rough, and impolite. Come and behold the difference between Jacob and Esau. Jacob talked to his father humbly, with humility. It is written, “And he came to his father, and said, My father” (Ibid. 18). The difference between the language of Esau and Jacob is that Jacob did not want to frighten him. Thus, he spoke humbly, saying “arise, I pray you, sit and eat of my venison.” Esau, however, said “Let my father arise,” as if he was not speaking to him.
Too much formality can also indicate lack of connection, especially in more intimate relationships. Sometimes, I wonder if even the respect and admiration we attribute to great Torah leaders also could subconsciously feed an avoidance. That is, they are so great, as we can discuss stories of their wondrous intellectual or ethical feats, it also is a free pass to avoid holding ourselves to that standard. If they are revered as super human, then us regular humans are exempt from that expectation .