I think the key similarities are hero worship and, closely tied to it, their use of propaganda. Many Germans never saw Hitler for what he was; they continued to "trust" that he would resurrect their hopes. This was as much due to his charistmatic leadership as it was to their need for redemption from the humiliation of WW I.
The use of propaganda was even more masterful. Just as history is "created" in 1984, so Hitler's propaganda machine run by Joseph Goebbles made sure that no information got to the German people that was critical of Hitler, the regieme or anything detrimental to their programs. Just as BB felt free to "change" the past, or what people thought to be the past (this is a very complicated issue), Hitler invented facts to support his ideology and politics. He made especially good use of the new medium of film ... and you have to wonder what he woujld have done if television were more widelyl available.
Of course, this goes on in our time as well, which is why the book is so frightening. It used to be difficult to find the truth because information was so hard to come by; there is now so much information on the internet that it's difficult/impossible for us to know what to believe. So politicians, especially in the election cycle, create realities for us. For example, we have recently been told that the policies of the present administration, while not seeming to work, prevented our entering a depression. How do know if this is true? How do they know if this is true? But it is convenient, which is kehy to BB's manipulation of the facts.
"The price of freedom is eternal vigilance."
In both settings, one sees how there is a lack of institutional control to limit the role of government. Both governments live and exist off of a blurring of the public and private, and an absence of limiting functions. They both thrive in the fact that they controlled every aspect of consciousness. Both used information and the means to achieve it in order to fully maximize their own power. Both governments needed to identify "scapegoats" in order to divert any form of questioning the authority structures. In the most sad of ways, both governments did an excellent job of ensuring that individuals betray others in order to identify dissenters and consolidate the power of the central government.
I can think of two comparisons right off -- one that is sort of a general one and one that is more specific.
The smaller, more specific similarity is between the Two Minutes Hate and German efforts to demonize the Jews. Of course, there was not literally a time each day for expressing hatred for Jews. However, the idea of the government trying to whip up hatred of some person or group of people is common to both the book and the historical era.
More broadly, the two societies are similar because the governments are trying to control the lives of the people and make them generally similar. The Nazis tried to remake everyone into what Hitler and his officials saw as perfect Germans. They used various groups (like the Spies and the Junior Anti-Sex League in the book) to help mold the people. They used surveillance and secret police to help enforce conformity.