Why was Simchat Torah instituted on Shemini Atzeret, and not on Shavuot, “the time of the giving of our Torah”? The standard answer is that “since we complete the Torah, it is fitting to rejoice upon its completion.”

Chassidut offers another answer-the rejoicing of Shemini Atzeret is for the second tablets, given on Yom Kippur, which were “doubled for strength,” i.e. having twice the spiritual value of the first ones. One can further explain Simchat Torah being connected to the second tablets, and not to the first ones, given on Shavuot:

1) The first tablets were “G‑d’s handiwork and G‑d’s writing,” symbolizing a gift from above. The latter tablets, by contrast, were G‑d’s writing on tablets made by Moses and thus symbolizing human effort. Simchat Torah is linked to the latter tablets since true rejoicing only comes through labor and effort.

2) The second tablets were given as a consequence of the Jewish people’s repentance, and therefore this bestowal in particular, marking the restoration of the Jewish people’s closeness with G‑d, inspires great rejoicing.

(Likkutei Sichot, vol. 14, p. 156; vol. 29, p. 232)