In the maamar in Likkutei Torah, the Alter Rebbe adds a further point: In addition to “receiving them all pleasantly,”וּמוֹסִיף בְּהַמַּאֲמָר
the king “shows a smiling countenance to all.”וּמַרְאֶה פָּנִים שׂוֹחֲקוֹת לְכֻלָּם,
The difference between “receiving (them all pleasantly)” and “showing (a smiling countenance)” can be explained as follows:דְּהַחִלּוּק בֵּין מְקַבֵּל (בְּסֵבֶר פָּנִים יָפוֹת) לְמַרְאֶה (פָּנִים שׂוֹחֲקוֹת) הוּא,
“Receiving,” מקבל, in the original, refers to accepting something that existed (before it was received);דִּלְשׁוֹן מְקַבֵּל נוֹפֵל עַל דָּבָר שֶׁיֶּשְׁנוֹ מִקֹּדֶם (לִפְנֵי שֶׁקִּבֵּל)
a person receives a preexisting entity.וְהוּא מְקַבֵּל אֶת הַדָּבָר,
“Showing a smiling countenance to all” indicates that one’s own “shining countenance”וּמַרְאֶה פָּנִים שׂוֹחֲקוֹת הוּא שֶׁהַפָּנִים שׂוֹחֲקוֹת שֶׁלּוֹ
(that he possessed even before he shows it)(שֶׁיֶּשְׁנָם גַּם לִפְנֵי שֶׁמַּרְאֶה)
radiates forth and is revealed to another person.הוּא מַרְאֶה וּמְגַלֶּה אוֹתָם לְזוּלָתוֹ.
On this basis, we can understand why after the maamar states that the king “receives them all pleasantly,”וְזֶהוּ דִּלְאַחֲרֵי שֶׁאוֹמֵר שֶׁהַמֶּלֶךְ מְקַבֵּל אֶת כֻּלָּם בְּסֵבֶר פָּנִים יָפוֹת
it adds that he “shows a smiling countenance to all.”מוֹסִיף שֶׁהוּא מַרְאֶה פָּנִים שׂוֹחֲקוֹת לְכֻלָּם,
By receiving them all pleasantly,דְּזֶה שֶׁהוּא מְקַבְּלָם בְּסֵבֶר פָּנִים יָפוֹת הוּא
the king shows that the desire of the people (to greet him) is accepted pleasantly by him.שֶׁהָרָצוֹן דְּהָעָם (לְהַקְבִּיל אֶת פְּנֵי הַמֶּלֶךְ) מִתְקַבֵּל אֵצֶל הַמֶּלֶךְ בְּסֵבֶר פָּנִים יָפוֹת,
This desire to greet him is what existed before and is now accepted by the king.
Adding that the king “shows a smiling countenance to all” indicatesוְהַהוֹסָפָה דְּמַרְאֶה פָּנִים שׂוֹחֲקוֹת הוּא
that when a person is aroused with a desire to turn to G‑d in teshuvah, a revelation of “the smiling countenance” Above is drawn down to him,דִּכְשֶׁמִּתְעוֹרֵר בְּרָצוֹן לַעֲשׂוֹת תְּשׁוּבָה נִמְשָׁךְ לוֹ הַגִּלּוּי דְּפָנִים שׂוֹחֲקוֹת דִּלְמַעְלָה,
i.e., he is shown the pleasure (reflected in the smile) of the King Himself.הַתַּעֲנוּג (שְׂחוֹק) דְּהַמֶּלֶךְ עַצְמוֹ.
Chassidus1 identifies taanug, pleasure, with the very core of the soul.
What is the source for that association? The soul’s other powers – emotion, intellect, and even will – express the soul’s energy as it extends itself outward and interacts with other people, situations, and concepts. By contrast, when the soul feels pleasure, it is turning inward and establishing contact with its core. The object that brings the soul pleasure is merely a catalyst that awakens the soul and enables the soul to more fully sense its own self.
Within this general definition, Chassidus2 distinguishes between taanug hamurkav, compounded pleasure, and taanug habilti murkav, simple pleasure. Whenever a person derives pleasure from a particular object or activity, it brings him into contact with his soul. The song, piece of art, or activity strikes a cord within him. It takes him beyond his everyday here and now and enables him to sense inner energies which often lie latent within his consciousness. However, since it is the object or the activity that awakens the connection with his soul, that object or activity diverts a certain dimension of the awareness from the soul as it is in its essence. This is the meaning of the term “compounded pleasure”; there is an awareness of something else aside from the soul. With regard to “simple pleasure,” by contrast, all that is felt is the soul.
The extent to which the soul’s pleasure is compounded depends on the nature of the catalyst for the pleasure. Take, for example, the pleasure that comes from a song: A deep and thoughtful song elicits an entirely different degree of pleasure than a light and happy song. The content and intensity of the deeper song add additional elements, aside from the inherent pleasure the soul experiences from it. The more substantial the catalyst, the greater the diversion from the focus on the soul’s essence.
This pleasure is far loftier than the pleasure (the King displays when “receiving them all pleasantly”),דְּתַעֲנוּג זֶה הוּא לְמַעְלָה מֵהַתַּעֲנוּג (דִּבְחִינַת פָּנִים יָפוֹת)
which is aroused by the teshuvah of the people.שֶׁמֵּהִתְעוֹרְרוּת הַתְּשׁוּבָה,
“Receiving them all pleasantly” reflects the pleasure the King takes as a result of the people’s coming out to greet Him, i.e., pleasure that results from an external factor. “Showing a smiling countenance to all” reflects the King’s inner pleasure, His innate delight that is not at all dependent on any external factor.
The term פָּנִים שׂוֹחֲקוֹת translated as “a smiling countenance” literally means “a laughing countenance.”
As explained in another source,3וְכַמְּבֹאָר בְּמָקוֹם אַחֵר
the source for the pleasure reflected in laughter (“a smiling countenance”) is the essence of pleasure,דְּשֹׁרֶשׁ הַשְּׂחוֹק (פָּנִים שׂוֹחֲקוֹת) הוּא בְּעַצְמוּת הַתַּעֲנוּג
(pure pleasure that is not compounded by any other influences or causes)(תַּעֲנוּג עַצְמִי הַבִּלְתִּי מֻרְכָּב)
which far surpasses pleasure that results from a given factor (pleasure that is compounded).שֶׁלְּמַעְלָה מִתַּעֲנוּג הַבָּא עַל יְדֵי דָּבָר (תַּעֲנוּג הַמֻּרְכָּב).
When a person laughs, he is uncontrolled; he loses consciousness of his self. And by doing so, he taps into the essence of his soul that is also unbounded.
It is possible to explain that the parallel to “the smiling countenance” in the analogueוְיֵשׁ לוֹמַר, דְּעִנְיַן פָּנִים שׂוֹחֲקוֹת בְּהַנִּמְשָׁל
is the pleasure G‑d takes in the Jewish people themselvesהוּא הַתַּעֲנוּג דִּלְמַעְלָה שֶׁבְּיִשְׂרָאֵל עַצְמָם
(which is loftier than the pleasure that results from the Jews’ observance of the Torah and its mitzvos(שֶׁלְּמַעְלָה מֵהַתַּעֲנוּג שֶׁמִּקִּיּוּם הַתּוֹרָה וּמִצְוֹת דְּיִשְׂרָאֵל,
and even loftier than the pleasure that results from the Divine service of teshuvah).וּלְמַעְלָה גַּם מֵהַתַּעֲנוּג שֶׁמֵּעֲבוֹדַת הַתְּשׁוּבָה)
The pleasure that G‑d derives from the Jews’ Divine service reflects pleasure that comes from an outside influence, while the pleasure which G‑d derives from the Jews themselves is an essential pleasure.
One of the reasons given for the advantage of baalei teshuvah (those who turn to G‑d in teshuvah) over the righteous is that teshuvah comes from the soul itself, which is above the Torah.4 Indeed, the very meaning of the term teshuvah, “return,” i.e., the soul returns to – and expresses – its very core, points to this concept. Therefore, through teshuvah, the essential quality of the Jews themselves is afterwards revealed.
Nevertheless, teshuvah is just a preparation for the revelation of the pleasure in the Jews themselves – for the soul itself surpasses even teshuvah.5 For that reason, G‑d’s pleasure in the Jews is aroused by their desire to turn to Him even though, as explained above, they have yet to be successful in overcoming the obstacles preventing them from turning to G‑d in complete teshuvah.
This pleasure lies within G‑d’s very Essence.דְּתַעֲנוּג זֶה הוּא בְּהָעַצְמוּת.
When the King “shows a smiling countenance” to the peopleוְעַל יְדֵי שֶׁמַּרְאֶה לָהֶם פָּנִים שׂוֹחֲקוֹת,
and this pleasure is revealed to a person (who is aroused with a desire to teshuvah),שֶׁתַּעֲנוּג זֶה מִתְגַּלֶּה לְהָאָדָם (הַמִּתְעוֹרֵר בְּרָצוֹן לִתְשׁוּבָה),
this awakens within himזֶה מְעוֹרֵר אֶצְלוֹ
(“as water reflects a face”)6(כַּמַּיִם הַפָּנִים לַפָּנִים)
pleasure in G‑dliness,הַתַּעֲנוּג בָּאֱלֹקוּת
so much so that the person’s pleasure in G‑dliness becomes the essence of his pleasure.וְעַד שֶׁהַתַּעֲנוּג בָּאֱלֹקוּת הוּא עֶצֶם הַתַּעֲנוּג שֶׁלּוֹ,
This is possible because G‑dliness lies at the very core of his being; the soul of every Jew is an actual part of G‑d.7
This gives him even more power to overcome any obstacles and hindrancesוְזֶה נוֹתֵן לוֹ הַכֹּחַ עוֹד יוֹתֵר לְהִתְגַּבֵּר עַל הַמְּנִיעוֹת וְעִכּוּבִים
and turn to G‑d in complete teshuvah.וְלָשׁוּב בִּתְשׁוּבָה שְׁלֵמָה.
Since the pleasure in G‑dliness that lies at the core of his being is consciously felt, nothing can prevent its expression.

Summary

The difference between “receiving them all pleasantly” and “showing a smiling countenance to all” can be explained as follows: “Receiving” refers to accepting something that existed before. “Showing a smiling countenance to all” indicates that one’s own “shining countenance” is radiated and revealed to another person.
By receiving all the people pleasantly, the king shows that he pleasantly accepts their desire to greet him. Adding that the king “shows a smiling countenance to all” indicates that when a person is aroused with a desire to turn to G‑d in teshuvah, a revelation of “the smiling countenance” from Above is drawn down to him. The pleasure reflected in a smile and laughter is the essence of pleasure that is not compounded by any other influences or causes.
It is possible to explain that the parallel to “the smiling countenance” in the analogue is the pleasure G‑d takes in the Jewish people themselves (which is loftier than the pleasure that results from the Jews’ observance of the Torah and its mitzvos). When G‑d “shows a smiling countenance” in Elul, this awakens pleasure in G‑dliness within a Jew and empowers him to overcome any obstacles and hindrances and turn to G‑d in complete teshuvah.