Gandalf the Grey and Gandalf the White
 

A question:

Why is it treated almost as if Gandalf the Grey and Gandalf the White are two different entities? Both are 'cloaks' for the immortal spirit of Olorin...Yet they are treated very differently. I'm talking more in terms of the films, in that Gandalf the Grey seems very different from Gandalf the White...I haven't read the books in ages so I'm wondering, is the characterization the same, and if so, why?

And why is Gandalf unable to remember that he was called Gandalf when he first arrived back?

When Gandalf returns Gimli himself inquires why the new color and Gandalf responds saying, "Indeed, I am Saruman, one might almost say, Saruman as he should have been." [TTT, pp. 116-117] Gandalf also does mention forgetfulness and a relearning process. Aragorn claimed that the White Rider is mightier than the 9 Ring Wraiths, although I'm not sure if earlier in book one Gandalf would agree to such a claim. Maybe he would then too.

In the book at least, Gandalf as the White has been given by his superiors in Valinor all the "authority" that previously had been Saruman's. Gandalf was then made the ultimate commander-in-chief of the war against Sauron, and that was recognized by Aragorn, Galadriel, Théoden, and all the allies of the West (excepting Denethor), whereas before he was just an "adviser".
I think the handover of power was made "official" when Gandalf informed Saruman he no longer had a color, and broke Saruman's staff.