The Koznitzer Maggid teaches that this "yom ha'geshamim," the day of rain the gemara refers to, is a hint to the day of Shmini Atzeres when we start saying mashiv ha'ruach u'morid ha'geshem. Our focus on this day is not only on the spiritual rain that we want to fall, but also on the "spiritual rain" we want to fall as well.
In reality Shmini Atzeres should be 50 days after Sukkos just as Atzeres (Shavuos) occurs 50 days after Pesach. But, says the Midrash Tanchuma, Hashem does not want us to leave our homes in the winter to make aliya la'regel, so he tacked on Shmini Atzeres right after the seven days of Sukkos. The association between Shmini Atzeres / Simchas Torah (in chutz la'aretz we seperate off Simchas Torah as an added day; in Eretz Yisrael there is only one day of Yom Yov) and Shavuos underscores the connection between this Yom Tov and the idea of mattan Torah. Hashem gave us Pesach even though we were bereft of mitzvos and zechuyos; it therefore took 7 weeks for us to earn the right to stand at Har Sinai. After a holiday of Sukkos replete with the mitzvos of sukkah, of lulav and esrog, we earn the right to Atezres in only 7 days. Sefas Emes writes that Shavuos is the holiday of Torah sheb'Ksav; Shmini Atezers is the Torah we have earned of our own accord, the holiday of Torah sheBa'al Peh.