When reading the Silmarillion it seems very clear to me that Dragons are much more powerful in battle than Balrogs. First there were many more Balrogs in the Battles of Beleriand than dragons. Before the War of Wrath there was only Glaurung and he alone seems to be more powerful than all the dozens or hundreds of Balrogs together.
Even when he was very young he devastated Ard Galen all alone and it needed many riders to stop him:

Of the Return of the Noldor said:

He was yet young and scarce half-grown, for long and slow is the life of the dragons, but the Elves fled before him to Ered Wethrin and Dorthonion in dismay; and he defiled the fields of Ard-galen. Then Fingon prince of Hithlum rode against him with archers on horseback, and hemmed him round with a ring of swift riders; and Glaurung could not endure their darts, being not yet come to his full armoury,

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Balrogs in opposite were more less of equal powerful as a Noldo:

Of the Return of the Noldor said:

Fëanor was surrounded, with few friends about him. Long he fought on, and undismayed, though he was wrapped in fire and wounded with many wounds; but at the last he was smitten to the ground by Gothmog, Lord of Balrogs, whom Ecthelion after slew in Gondolin.

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During the Dagor Bragollach you can easily comprare the effectiveness of the Balrogs with full grown Glaurung:

Of the Ruin of Beleriand and the Fall of Fingolfin said:

by the valour of the Elves and the Men of the North, which neither Orc nor Balrog could yet overcome, Hithlum remained unconquered,

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Of the Ruin of Beleriand and the Fall of Fingolfin said:

But they overwhelmed the riders of the people of Fëanor upon Lothlann, for Glaurung came thither, and passed through Maglor's Gap, and destroyed all the land between the arms of Gelion.

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And of course there is the War of Wrath were the Valar easily destroyed Morgoth's hosts:

Of the Voyage of Eärendil and the War of Wrath said:

There was marshalled the whole power of the Throne of Morgoth, and it had become great beyond count, so that Anfauglith could not contain it; and all the North was aflame with war. But it availed him not. The Balrogs were destroyed, save some few that fled and hid themselves in caverns inaccessible at the roots of the earth; and the uncounted legions of the Ores perished like straw in a great fire, or were swept like shrivelled leaves before a burning wind

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But then Morgoth released the dragons:

Of the Voyage of Eärendil and the War of Wrath said:

But he loosed upon his foes the last desperate assault that he had prepared, and out of the pits of Angband there issued the winged dragons, that had not before been seen; and so sudden and ruinous was the onset of that dreadful fleet that the host of the Valar was driven back, for the coming of the dragons was with great thunder, and lightning, and a tempest of fire.

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After the first surprise the Valar together with the eagles and Earendil destroyed the dragons too, but the difference with the balrogs is striking.
Now I already hear some of you saying: "But the Dragons of the War of Wrath wre much larger than Smaug!" I don't think so. I believe Smaug was present at the War of Wrath! Why? Some Dragons escaped the destruction:

Of the Voyage of Eärendil and the War of Wrath said:

and well-nigh all the dragons were destroyed

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. As I understand it, these were the later Dragons of the North with Smaug the greatest among them. I don't know any direct statement by Tolkien about this. But I also don't see any hint that dragons reproduce. So it seems the most obvious origin of those dragons.

I found a quote that dragons bred indeed:

LotR - Appendix A - Durin's Folk said:

But there were dragons in the wastes beyond; and after many years they became strong again and multiplied, ...

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So it's unsure if Smaug was born after the end of First Age. But as he was "greatest of the dragons of his day" I would anyway think him comparable to the dragons of the War of Wrath.

Another question is if the Balrog in Moria was more powererful than the Balrogs of Beleriand. The answer is that it was acutally one of them:

LotR - Appendix A - Durin's Folk said:

Thus they roused from sleep a thing of terror that, flying from Thangorodrim, had lain hidden at the foundations of the earth since the coming of the Host of the West: a Balrog of Morgoth.

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The Balrog of Moria was slain by Gandalf but Gandalf died too. So was it of equal power as a Maiar? Yes, Balrogs are actually Maiar:

Valarquenta - Of the Enemies said:

For of the Maiar many were drawn to his splendour in the days of his greatness, and remained in that allegiance down into his darkness; and others he corrupted afterwards to his service with lies and treacherous gifts. Dreadful among these spirits were the Valaraukar, the scourges of fire that in Middle-earth were called the Balrogs, demons of terror.

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So the Balrogs were not even the least powerful among the Maiar who served Morgoth. The power of Maiar varies extremly.
Gandalf on the other hand was not as powerful as he was in Valinor during that battle. The Istari were not allowed to use their powers:

Unfinished Tales - Part IV -The Istari said:

For with the consent of Eru they sent members of their own high order, but clad in bodies of as of Men, real and not feigned, but subject to the fears and pains and weariness of earth, able to hunger and thirst and be slain; though because of their noble spirits they did not die, and aged only by the cares and labours of many long years. And this the Valar did, desiring to amend the errors of old, especially that they had attempted to guard and seclude the Eldar by their own might and glory fully
revealed; whereas now their emissaries were forbidden to reveal themselves in forms of majesty, or to seek to rule the wills of Men and Elves by open display of power, but coming in shapes weak and humble

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Even in Valinor Gandalf (Olorin) wasn't a very powerful Maia:

Unfinished Tales - Part IV -The Istari said:

But Olórin declared that he was too weak for such a task, and that he feared Sauron. Then Manwë said that that was all the more reason why he should go, and that he commanded Olórin

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So I'd say Gandalf when he battled the Balrod was no more powerful than a Noldo during First Age and the Balrog of Moria no more powerful than the Balrogs of Beleriand.