So, when Gandalf agreed with Frodo's decision to travel through Moria, he knew that something could be hiding there. Yet, he had no reason to think that it would be something as awful and powerful as a Balrog. Only King Dain Ironfoot knew that Durin's Bane was a Balrog.
However, in the Tolkien novel, Gandalf actually never knew about the Balrog. All he knew was that Moria had been evacuated because of something called Durin's Bane, and that it still lurks inside.
The company fled with Gandalf, but the Orcs and the Balrog, taking a different route, caught up with them at the Bridge of Khazad-dûm. At the Bridge, Legolas instantly recognized it as a Balrog, and Gimli recognized it as Durin's Bane.
The Balrog killed Durin VI, the Dwarf-King of Khazad-dûm, whereafter it was called Durin's Bane by the Dwarves. Avarice, principally for mithril, drove the dwarves to go too deep and awaken the Balrog.
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.
So, when Gandalf agreed with Frodo's decision to travel through Moria, he knew that something could be hiding there. Yet, he had no reason to think that it would be something as awful and powerful as a Balrog. Only King Dain Ironfoot knew that Durin's Bane was a Balrog.
The fear of Elves is more spiritual, so the demonic creatures of shadow and flame are unsurprisingly horrifying to them. Legolas doesn't appear to have any hesitation in his actions throughout the films, but it's clear that his blood runs cold when a Balrog appears in the Mines of Moria.
But I think you answered your own question. Gandalf doesn't know what it is until he sees it out in the open, as Legolas does as well. He recognized what it is as well as the elf, by sight, but is unable to recognize it just by confronting its power with his own.
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.
Long it stayed beneath the realm of Khazad-dûm, until the Dwarves there mined too deeply for mithril and unearthed the Balrog of Morgoth. The creature slew their king, Durin VI, earning the name "Durin's Bane", drove the Dwarves out and claimed Khazad-dûm (now called Moria) as its domain.
In the end Tolkien stated that there were probably "at most" seven Balrogs: In the margin my father wrote: 'There should not be supposed more than say 3 or at most 7 ever existed. '
There had been no communication from Moria for quite some time, but no one knew exactly what had happened there, or why the colony had gone silent. Gandalf definitely did not know about the Balrog, or even about the orc attacks that had sent Balin and all who were with him to an early grave.
Still, the dragons proved more effective in battle. Then there's Gandalf the Grey, who managed to kill a Balrog single-handed, but decided to manipulate some Dwarves and one very reluctant hobbit into dealing with Smaug.
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.
He pursued his foe for days through dark tunnels, and then all the way up the Endless Stair to the top of Durin's Tower in the living rock of Zirakzigil, where the two engaged in a two-day long battle at the pinnacle of the Silvertine above the clouds.
Sauron's deepest fear was that Aragorn would use the Ring against him. With its powers of domination, Aragorn could comfortably take over the entirety of Sauron's dark army. Though this would eventually corrupt Aragorn, it would allow him to destroy Sauron and take his place.
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.
In general Sauron did not seem to be afraid of the Balrog during the Third Age. The behavior of Durin's Bane indicated that he would not leave Moria and its tunnels : he was not a rival for the control of Middle-earth and the Balrog did not share Sauron's abilities as a potential Dark lord.
Combining the poor communication between Erebor and Moria with the Dwarves' extended lifespans, Gimli might not have known about the fall of Moria simply because Erebor had not grown too worried about the lack of information yet.
Smaug could easily have cut through the armies of Mordor, and delayed the invasion of Middle-earth by years. So it wasn't in Sauron's interest to take on the dragon, but that doesn't mean he saw it as a lost cause, as he fully planned on creating an alliance.
