So, in terms of "angelic rankings," they were on the same level, which meant that Sauron had no authority to commander the Balrog. As a Valar, Morgoth led a whole slew of Balrogs in the First Age, but because Sauron was only a Maia, the Balrog would have had no loyalty to him.
The Balrogs were originally the servants of Morgoth in the First Age, of whom, Sauron was also his servant. The Balrogs and Sauron were all Maiar and somewhat coeval; it is debatable whether a Balrog would recognize Sauron as successor and serve him willingly.
Secondly, Balrogs were the epitome of malice and power while Sauron only ever won one of three fights he ever had. Sauron was the great deceiver. His power lies in planning, subterfuge, manipulation, and corruption. Strength wasn't his game.
Balrogs, also called Valaraukar, were originally Ainur created by Ilúvatar, probably those who joined Melkor during his discordance in the Music of the Ainur. After entering into Eä, they were Maiar, lesser spirits at the service of the Valar.
Why Didn't Sauron Use the Balrog? Middle-Earth Lore
Who killed 5 Balrogs?
There is no account of any dwarf ever killing a Balrog, let alone multiple. The only recorded instances of Balrogs having been slain are Gandalf's slay of Durin's Bane, Echtelion's slay of Gothmog and Glorfindel's slay of an unnamed Balrog.
The Elf Ecthelion leads the charge against the Orcs, and fights Gothmog, the greatest Balrog; they wound each other and both fall into the king's fountain in Gondolin; both drown.
In all of Tolkien's text I've read, there are three people who have killed a Balrog. Ecthellion, Glorfindel and of course Gandalf. All of these fights ended in a draw with all three sacrificing their lives for the deed.
Sauron does not fear Durin's Bane because essentially they had been allies back “in the olden days". Both Balrogs (all balrogs) and Sauron chose to serve Melkor/Morgoth.
From a technical sense, Sauron and the Balrogs were Maiar, not Valar. That meant that they were designed as lesser beings from the beginning of time. Thus, in their original forms, Morgoth would have easily defeated Sauron in a fight.
Morgoth, the first to reign terror upon Middle-earth, was more powerful than Sauron and commanded an army of dragons and Balrogs. While both Morgoth and Sauron were cunning, Morgoth's rule and terror lasted longer and his corruption spread further, making him more formidable in the Lord of the Rings universe.
As a Valar, Morgoth led a whole slew of Balrogs in the First Age, but because Sauron was only a Maia, the Balrog would have had no loyalty to him. He was his own master in Moria, and no Dark Lord was going to ruin that.
He will fight in the Last Battle against the Valar and their allies, but will ultimately be slain by Túrin Turambar, the Man he cursed. By finally defeating Morgoth, Túrin will avenge not only himself, but all members of the race of Men.
If Balrogs are corrupted Maiar then would Saruman eventually become a Balrog? No, balrogs were specific maiar that chose that particular physical shape, liked it and stayed that way therefore locking themselves permanently in that shape.
Sauron served Aulë, the smith of the Valar, acquiring much knowledge; he was at first called Mairon ("The Admirable", in Tolkien's invented language of Quenya) until he joined Melkor. In Beleriand, he was called Gorthu "Mist of Fear" and Gorthaur "The Cruel" in Sindarin, another of Tolkien's invented languages.
Despite her power, Galadriel couldn't match Sauron's ability to wield the One Ring, and it would have ultimately turned against her. Even if she managed to carry the ring, Galadriel's ambition and power-seeking nature would have eventually led to her corruption, making her unfit to deal with Sauron's weapon.
At some point prior to The Rings of Power, Sauron conjured a fresh physical form. He then seized the Southlands' crest from its last known king (who died 1000 years prior to the TV show's timeline) and assumed the identity "Halbrand."
For more than five millennia, the Balrog hibernated in his deep hiding place at the roots of the mountains in Khazad-dûm. He remained undisturbed throughout the Second Age and most of the Third, before the mithril-miners of dwarf-King Durin VI awoke him in T.A. 1980.
Oh, yes, there are creatures more frightful than Balrogs in Middle Earth. Ungoliant who was able to cry havoc in all Valinor is easily more terrifying than Balrogs, even Morgoth feared its gigantic and hideous form. It took a group of Balrogs and Morgoth himself resisting Ungoliant's attack to make her flee.
Summary. Gandalf's main reason for supporting Thorin's quest in The Hobbit was to eliminate the threat of the dragon Smaug, who would be a powerful ally for Sauron. Despite popular belief, the Balrog known as Durin's Bane was more powerful than Smaug in Middle-earth, being a literal, primordial demon.
Balrog vs dragon, as a general setup, is simple to answer. Balrog. Even in the earliest writings, when Balrogs were much, much weaker and more numerous than they are in the conceptions you probably identify with them (as in, what Gandalf fights), Balrogs are stated outright to be more powerful than dragons.
