After his fight with the Balrog of Moria, Gandalf changed:
Gandalf was sent back as a significantly more powerful figure; Gandalf the White.
- "Gandalf - Gandalf the White", Wikipedia
What are Gandalf's new powers as Gandalf the White? What feats could he accomplish that he was not able to perform as Gandalf the Grey?
Eru, the Authority, sent back Gandalf with additional power, knowledge and wisdom.
He was sent by a mere prudent plan of the angelic Valar or governors; but Authority had taken up this plan and enlarged it, at the moment of its failure. 'Naked I was sent back – for a brief time, until my task is done'. Sent back by whom, and whence? Not by the 'gods' whose business is only with this embodied world and its time; for he passed 'out of thought and time'.
Prior to Gandalf's return, he would not have been able to rescue Theoden by breaking Saruman's spell and inspiring and leading the Rohirrim. Nor would he have been able to face down Saruman, as originally Saruman had been the leader of the Istari and hence more powerful than Gandalf.
So Gandalf sacrificed himself, was accepted, and enhanced, and returned. 'Yes, that was the name. I was Gandalf.' Of course, he remains similar in personality and idiosyncrasy, but both his wisdom and power are much greater. When he speaks he commands attention; the old Gandalf could not have dealt so with Theoden, nor with Saruman. He is still under the obligation of concealing his power and of teaching rather than forcing or dominating wills, but where the physical powers of the Enemy are too great for the good will of the opposers to be effective he can act in emergency as an 'angel'
Gandalf's barring of the Witch-King from Minas Tirith is another example. Previously, he merely aided Elrond's flood in keeping the Nazgul from Rivendell. After his return he was able to single-handedly impede the Witch-King.
He alone is left to forbid the entrance of the Lord of Nazgûl to Minas Tirith, when the City has been overthrown and its Gates destroyed
Excerpts from Letter 156.
Adding to @dlanod answer.
When I read the books the main feature that seemed to stand out the most as Gandalf The White, was his confidence on himself.
As The Grey he underestimated his powers, he never overdid it and he was VERY humble (this was one of the main reasons that the Valar sent him and the main reason he received Narya). As The White he still knows his limits but is far more daring and defying towards Saruman and Sauron (he frees Theoden and casts Saruman from the Order, he faces the Witch King, etc)
Gandalf the white is more powerful because the whole story is about the fall. The fall from grace, wisdon, loyalty...anything. Anything that sets the creator apart from the sub-creators, ie-Eru from Valar, Maiar, elves, men, etc. All the main characters experience a fall of some kind, some repent some don't, thus we observe the duality of good vs evil as polar expressions of divinity. This is a recurring theme. As Gandalf's "fall" is based on a sacrifice to the higher good (Eru), his re-embodiment is a divine blessing for the benefit of the sub-creators. He cannot conquer, only inspire. He cannot force anything to his will but he can use force to preserve the will of others. He is, in a sence, Eru! The Valar and Maiar are "the offspring of the thought of illuvitar". Which we can take to mean, aspects of the divine self. Gandalf the White is God, embodied and Sauron is mearly dancing to the divine tune. Gandalf's greatest power is his ability to know how Sauron will react. If you knew all the moves your opponent would make beforehand, could you really lose at chess? No. The overall message is the choice of the sub-creators to follow the wisdom of the creator. Frodo is the hero who trusted, had faith in, Gandalf. As is Sam, Aragorn, Theoden and even Butterbur. Who trusted Sauron?
Gandalf is overall more powerful as Gandalf the White. He is far more charismatic and has some limited foresight. He was able to defeat Saruman, breaking Saruman's power over Theoden and breaking his staff after the Ents took Isenguard. However, he was not all-powerful, seeing as the Witch-King of Angmar broke Gandalf's staff. Still, he is obviously more powerful, since, as Gandalf the Grey, he was defeated by Saruman, then he later broke his staff and cast him from the Istari and the White Council.
Gandalf was more powerful as white than grey. His wisdom as well as his magical powers and his physical powers were all increased. He basically was now Saruman (like he said). He was the new White Wizard and was as powerful as Saruman was. Saruman lost a great deal of his power and was cast from the order.
Gandalf was sent back by , I assume, the Valar, who sent him in the first place. This is because he had not yet finished his task, which was to oppose Sauron and save Middle-Earth from him. The fact that he was sent back as a white wizard, instead of a gray, would tell you something if you had done your research. The "wizards", or Istari, are ranked by color of their robes. Saruman has white robes because he is the most powerful. Gandalf has grey, because he is less powerful, but still one of the most powerful, possibly the second most powerful of the five. When he was sent back to Middle-Earth as a white Istari, he gained a significant amount of power that came with the rank, or the other way around, however you see it. So, it is expected that he would be more powerful after his return.
