How Gandalf The Grey is Different from Gandalf The White in Lord of the Rings
Before becoming the Lord of the Rings Gandalf the Grey had to change more than just the attire of Gandalf the White. Through the writings of J.R.R. Tolkien we can understand the change of Gandalf. During the Fellowship of the Ring Gandalf fought a war against Balrog after which his appearance changed entirely. He was presented before the fellowship and denied to be Gandalf the Grey but claimed that he is Gandalf the White. Now let's just understand how Gandalf the white is different from Gandalf the Grey.
To get the whole idea first we need to know about Gandalf's birth childhood and history. He was from the Maiar People who were responsible for the emergence of Arda (the middle location of Earth). He carried such great strength that he performed the role of a divine protector. He was on a mission to defeat Sauron. He wandered in Middle-Earth for centuries wearing his wizard attire. With the passage of time he learnt a lot about Middle-Earth and also got to know about Sauron's return. Thereafter he along with Bilbo Baggins and his nephew Frodo found and dismantled one of the rings.
During the Fellowship to help Frodo in dismantling the One Ring Gandalf was led to Mines of Moria and died there but that was not the end of Gandalf. He returned to The Two Towers in an entirely different appearance as Gandalf The White and with more strength. It was a saying that the God Eru the supreme God of Arda had sent him back to accomplish his mission. His new appearance was all white including his clothes and hair. But there was something more that can be observed.
Gandalf the white had received powers from the Valar the Creation God of Adar. Afterwards he became alert and wiser. Actor Ian McKellen described Gandalf the white as more serious and less humble as Gandalf the Grey was. This time he didn't have time to spare with Bilbo Baggins and Hobbits. His White appearance indicated his higher place in the agents of the Valar. Before Gandalf Saruman was the head of agents of the Valar but he deceived the divine powers and joined Sauron. Now the power to punish Saruman was in the hands of Gandalf the White.
After getting more powers Gandalf the white became harsher than ever before. This can be proved by the incident of The Two Towers where he had immediately thrown out the malice King Théoden. Another Example Gandalf the white had destroyed Saruman's staff despite losing to him. At the time of the arrival of the King he was seen with his upgraded staff and sword. Gandalf the white had new skills which included talking to his horse Shadowfax and enhanced mental wisdom. No doubt Gandalf the White became more powerful and wiser but he had lost the charm that was in Gandalf the Grey. He made it to the strongest Fellowship of Rings but had lost the appeal of Gandalf the Grey during the Battle against Sauron.