Looking at which of these creatures was the 'greater evil' (which to my mind means which was more physically/sorcerously powerful), I am inclined to go with dragons, although I am by no means positive.
I do not think that Narya's not being destroyed by the Balrog of Moria is any real indication that Balrog-fire is less 'powerful' than Dragon-fire, since, as you said, Gandalf may not have even had the Ring with him, & he also may not have been directly hit with any huge blasts of Balrog-fire. I think a better indication of the superiority in power of Dragon-fire is Gandalf's saying in "The Shadow of the Past" that:
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It has been said that dragon-fire could melt and consume the Rings Of Power, but there is not now any dragon left on earth in which the old fire is hot enough; nor was there any dragon, not even Ancalagon the Black, who could have harmed the One Ring, the Ruling Ring...
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while failing to mention anything about the destructive capabilities of Balrog-fire.
As you have already established, there were Balrogs marching in the host of Glaurung, a possible indication of their inferiority. While one might make the argument that the Balrog of Moria in
The Lord of the Rings seemed more powerful than Smaug in
The Hobbit, this seems mostly attributable to the fact that J.R.R. Tolkien did not intend for Smaug to be considered among the dragons of the First Age like Ancalagon & Glaurung, as the characters & events of
The Hobbit were initially intended to be no more canonical to the history of Middle-earth than the dragon Crysophylax in
Farmer Giles of Ham or the land of Faery in
Smith of Wooton Major.
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...dragons are very clever and capable of speech. It is not said whether any balrog has either of these qualities.
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Very true. What made Glaurung so powerful was not only his brute strength & his ability to breathe fire, but also his mastery of hypnotism and persuasion. There is reason to believe that dragons were more powerful than Balrogs because of the
Silmarillion description of Glaurung's ability to hypnotize ('dragon-spell') &, to a high degree, control the minds of his victims as he saw fit. Smaug also shows a similar (though certainly lesser) mastery of psychological persuasion. Although one could make the argument that the Moria Balrog's spell-casting battle with Gandalf demonstrated a high level of (for lack of a better word) metaphysical ability, the Balrog's spells did not seem to hold anywhere near the same force as Glaurung's.
The awesome power of the dragons (or perhaps I should say 'some/many of the dragons') is exemplified in the emergence of Ancalagon & the winged dragons in the War of Wrath, described thusly in
The Silmarillion:
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...so sudden and so ruinous was the onset of that dreadful fleet that the host of the Valar were driven back, for the coming of the dragons was with great thunder, and lightning, and a tempest of fire.
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Never is anywhere near the same level of power ascribed to the hosts of Balrogs in
The Silm, nor can it ever be inferred from other descriptions of Balrogs in battle that Balrogs could come anywhere close to driving back the host of the Valar the way the winged dragons did. The Balrog of Moria could not even dispatch one Maia, a Maia whose power was extremely restricted at the time of the battle.
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Balrogs are definitely maiar, but dragons may be maiar too
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While some of the Balrogs definitely are Maiar, it seems at least somewhat debatable as to whether all the creatures that took a 'balrog' form in the First Age were Ainur in origin. In the
One hand tied behind their backs thread,
Heren-Istarion offers up the notion that some balrogs (as opposed to Balrogs) that marched into battle may have been:
- altered hroar of orks.
- altered hroar of Eruhini (Children of Ilúvatar)
as well as Maiar spirits initially corrupted to the service of Melkor.
That last part didn't really have anything to do with the 'greater evil' debate, just thought I'd throw it in there because I found it interesting, and because it relates to part of your post.
Unless some solid claim can be found to support the idea that Balrogs were stronger than dragons in Middle-earth, I am inclined to believe based on evidence from a variety of Tolkien's works that dragons were the mightier of the two.
-Númenor