Good topic, Brian.
I find myself agreeing very much with Ex.
It may seem nitpicking, but I cannot agree with calling any of the Ainur gods. The Ainur are the first creations of Eru (Tolkien's uncreated, eternal, omnipotent creator). Ainur is the plural form of Ainu (holy one). I prefer to call them angels rather than gods. Their powers are varied. They were created to enjoy Eru's presence and to assist in building his universe. Eru imparted to each of them gifts of knowledge and ability with which they were to engage in beautifying and maintaining Creation. To us, holy means sacred, hallowed, religious, or pious, but something or someone is sacred because it is different and set aside for a special purpose. The Ainur were not like Elves, Dwarves, Men, Ents, or Hobbits... they were different, set aside for Eru's special plans.
The first and mightiest of the Ainur are Manwe and Melkor (also called Morgoth). Manwe was gifted more of the mind of Eru than any other of the Ainur, but Melkor was gifted some of the individual knowledge that each Ainu was given. As Eru described his designs to the assembled Ainur, Melkor conceived a desire to begin his own creation. Since Melkor was born of the very thought of Eru and had been entrusted with a vast breadth of knowledge of Eru's plans, this was rebellion. Eru gently reined in Melkor, but Melkor fought back. Even after Eru rebuked Melkor and incorporated Melkor's flawed creations into his own, Melkor harbored resentment.
What did Melkor want? Melkor wanted to rule. Melkor, a created being who was given a personal vision and power from his creator, thought he could do a better job than his creator.
If you read the link in Extollager's post #7, you might compare Eru's treatment of Melkor with Mr. Biddle's treatment of Miss Sacco. I'd say Eru went very easy on treason.
What did Sauron actually want? He wanted to be like Melkor, his mentor.
Eru sent a number of the Ainur to Arda (the world in which Middle-earth exists). The fourteen mightiest were to lead the hundreds, perhaps thousands of of Ainur. The fourteen, led by Manwe, are the Valar and the rest are the Maiar. Melkor also went to Arda where he persuaded many Maiar to join him in opposing Manwe's efforts to fulfill the will of Eru. Initially, Melkor's opposition was merely the thwarting the environmental efforts of Manwe's group... this later became open war. One of the mightier Maia to join Melkor was Sauron.
He originally served Aule, the smith of the Valar. Sauron was probably instrumental in founding and maintaining Melkor's second greatest stronghold, Angband. The war between the Valar and Melkor lasted ages... and in the end they basically split the world into their two spheres of influence... Valinor (the land of the Valar) and Middle-earth. During this time, the Ainur were the only speaking and thinking creatures in Arda. Once the Valar became aware of the existence of Elves in Middle-earth, they committed themselves to total war against Melkor and defeated him. Melkor was imprisoned. I don't know what happened to all of the Maiar who served Melkor... I suspect some were forced to go to Valinor to be judged, I suspect some repented, but Sauron and the Balrogs escaped justice and went into hiding. When Melkor escaped like Napoleon from Elba, Sauron quickly rejoined him as his chief lieutenant. Sauron oversaw the destruction of Minas Tirith (no, not the same Minas Tirith... this one was a stronghold of the Elves of the First Age) and pillaging of Dorthonion (an Elven land that was crucial to encircling Angband).
When the Valar finally came forth to defeat Melkor for the final time, Sauron was captured. The Silmarillion says that Sauron seemed completely sincere in his confession of his guilt. He repented to Manwe's heralds, but when told that he'd have to go to Valinor to be personally judged by Manwe, Sauron fled. He escaped justice a second time and remained in Middle-earth to nurse his grudge against the Valar, the Eldar (the Elves, specifically the Noldor) and the Edain (the Men of the West).
In no way did the Valar reject his services. Sauron willingly joined Melkor's rebellion. Sauron willingly joined Melkor's second reign as Dark Lord. Sauron declined to answer to Manwe's judgement for thousands of years of rebellion, murder, and torture. I don't see any way that Sauron was a victim of the Valar nor was there ever any offer by Sauron to assist the Valar in protecting or healing the people and the land of Arda.
Unlike Melkor, Sauron's knowledge, passions, and powers lay only in smithing... well, and in murder, betrayal, and destruction, i.e. the things Melkor taught him. Sauron applied his knowledge of smithing to forge the One Ring... a master ring to control the nineteen rings of power made by the Elves. Three of these rings were made in secret and were the greatest of the Elven rings... Sauron never touched them.
If Sauron had won, what would have changed in Middle Earth? This is an easier question to answer.
Forty-four hundred years before Bilbo's party (give or take a few hundred), Sauron took the sixteen other rings of power made by the Elves. His force of Orcs crushed the Elves. The next fifteen hundred years or so were the Dark Ages of Middle-earth.
Sauron gave nine of those rings to human kings and lords.... these unfortunate men were turned into wraiths... Ring Wraiths. The Nazgul.
Three thousand years before Sam married Rosie, the last king of Numenor, Ar-Pharazon launched a campaign to wrest control of Middle-earth from Sauron. The Numenoreans were so numerous and so much better equipped that Sauron's armies fled. Sauron was imprisoned by Men and taken as captive to Numenor (home of the Men of the West, the progenitors of the Gondorians). Within two years, Sauron had moved from the dungeon to the palace as Ar-Pharazon's chief counselor. Within ten years, Sauron had the king and the court worshiping Melkor while he performed human sacrifices. And finally, Sauron convinced Ar-Pharazon that the Valar did not possess power in and of themselves... he lied that the land gave immortality to whomever ruled it... and Ar-Pharazon led his armies to assault Valinor itself.
Sauron wanted to corrupt Eru's creation and to thwart Manwe's plans because he liked controlling others. His rule would have included devil worship, human sacrifice, and eternal torture.
As an angel, a creature who has actually been in the presence of his creator, Sauron's continued disobedience to Eru's specific plan for him is treason.