In Israel Simchat Torah (Joy of the Torah) is celebrated on the day of Shmini Atzeret, and outside the land of Israel a day after. This shows that they are not the same holidays and both have their own importance.
Although a distinct holiday, Shmini Atzeret falls on the eighth day of Sukkot. Because the number 7 represents the natural world, we find that on all seven days of Sukkot, Jewish priests sacrificed bulls in the Holy Temple for all non-Jewish nations of the world.
Only one bull was slaughtered on Shmini Atzeret, the eighth day, which signifies time outside of the natural order. This one was reserved for Jews. As G-d’s people, they exist outside of the natural order of the world and have a direct spiritual relationship with G-d. Consider Sukkot to be a grand feast attended by G-d, Israel, and all non-Jewish nations. After the festival, when all the guests/people have left, G-d desires one more day of celebration with His beloved nation. This indicates that it is not appropriate for Noahides to celebrate this holiday.
We can read in “The Order”:
Althoug Sukkot is the time for judgemnet upen water, the custom of the Jewish world is not to recite the prayer for rain until after the holiday of Sukkot. It is appropriate for Noahides and Jews to pray for ain at the same time so that their prayers may join together in beseeching the Creator.
Do not withold water on our account – water, whose drops crown the tundra and grasses of the valley – but, rather, recall the merit of the righteous.
May He command His angel (Af-Bri) to apportion for us from the segragated rains, to soften the surface of the wasteland when it is as dry as a rock. Sooth those into whom a soul was blown with its drops, sustaining and enlivening those who recall the strong rains. Remember the righteous, whom You have blessed like a tree planted by streams of water, and whom You rescued from fire and water. Do not withhold water from the soil for their sake! Remember the righteous who, like water, pour out their souls before You. Give us abundance of water for the sake of their riggtheousness! Because You are the L-d G-d, Who causes the wind to blow and the rain to fall. Not for curse, but for blessing! For the sake of life, not death. In abundance, and not in scarcity. He is the One who sustains the living in kindness, supports the fallen, heals the sick, and releases those who are bound. Who is like You, the Master of Great Deeds, and the King who brings death and restores life, who makes salvation flourisch! Blessed are You, Lord, creator and sustainer of His creations! 
Simchat Torah is a Jewish holiday that appears to have originated in the 14th century. It commemorates the renewal of the weekly cycle of Torah readings. Many Noahides read and learn about the weekly chapters of Torah and thus feel instantly connected to Simchat Torah.
The day of Simchat Torah is an excellent day to remember that Moses received the full Torah from G-d at Sinai, including the Written and Oral Torah. G-d also repeated and handed Moses the Seven Commandments for the Noahides, complete with their explanations and specifics, in the Torah.
Moses painstakingly authored a Torah scroll (Deutr. 31:9), as well as one for each tribe, and these scrolls were subsequently correctly reproduced, so that thousands of years later we can still read this Word of G-d. That in itself is a huge miracle, and it brings us great joy to know that it allows us to read, learn, and study the proper way to live, as well as how to obey and serve G-d. After all, that is the entire reason of the world’s creation, as well as G-d’s purpose for our being here on Earth. To serve Him in the way He desires as King. Not because He requires it in order to be recognized as King, but because that is how
The Oral Torah, in addition to the Written Torah, brings us joy. Through its explanations, the Oral Torah clarifies numerous topics in the Written Torah. The Oral Torah was passed down to Moses, who in turn passed it down to Joshua, who in turn passed it down to today’s Orthodox rabbis, from whom we study.
The practical arrangement for celebrating Simchat Torah may include a festive meal with family and friends.
Experiencing the reading of the cycle of Torah by reading the last Torah-Portion of Deuteronomy followed by the first of Genesis.
The most striking feature of Simchat Torah is dancing with the closed Torah scroll. (Closed for the practical reason that it makes it easier to dance with, but also symbolically that the Torah is for everyone, including those for whom the Torah is “still” closed and “not yet” understood). For many, it will not be possible to live and experience this in a synagogue.
But enjoying beautiful songs, possibly with dancing, is of course possible.
By Angelique Sijbolts
Sources: he Divine Code by rabbi Moshe Weiner 4e edition p 25 end 38, Aish Article; Shmini Azeret and Simchat Torah, Talmud – Sukkot 55b, Recommended article: The Authenticity of the Written Torah
 The prayer in “The Order” is written for a communal function rather than a personal or individual prayer. Because not everyone will have the opportunity to say it as a communal prayer, I have somewhat modified it here so that it can be spoken as a personal – individual prayer. It is beneficial when you look at the prayer in its original form. See 
Youtube information Get Up for Simchat Torah – Roar Parody
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