The fact that the Tsarevich was sick was not announced publicly as portrayed in the movie. It was kept a secret.
Although considered to be a mystic, Rasputin was neither a monk and nor was he unmarried. He had left behind a wife and several children in his native village on the outskirts of Russia.
Paul is surprised to hear that the tsarevich suffers from hemophilia, and the doctor acts as if this is a recent problem. In fact, this was known more than ten years earlier, as soon as the child was born.
In the movie Rasputin is showed how he was giving the Tsar advices to go to WWI. However in real life he wasn't giving these advices during World War I.
The temperature of the tsarevitch is given in the daily bulletin as 96 degrees, on the Fahrenheit scale, which is used in the USA. However, in Russia the Celsius scale would have been used.
Natashia picks up a chandelier which casts a large shadow of her head up above her. BUT the lights on the candle were above her forehead. In order for these lights to have cast that shadow above her head they would have had to have been below her head, NOT above her head.
At the beginning of the movie it is told that Sergei Alexandrovich, Grand Duke of Russia, is killed. However this happened in 1905, 8 years before the scene's setting.
The time is 1914, but the women wear backless evening dresses in the style of 1932.
On the bulletin about the Zsarevitch's situation he is named by his real name in Russian but in the English translation he is simply called "Tsarevitch"
The Revolution is depicted in a montage including a shot of people running across an open area with the lower potion of Berlin's Brandenburg Gate in the upper-left background.
Captions identify St. Petersburg's districts (like Peterhof) being outside the city. They are technically part (and a district) of the big city.
When Prince Paul walks out the back door into the snow, dragging Rasputin with him, he turns around and closes and locks the door behind him. It does not seem likely that a mansion door that opens directly onto the pavement would have a key sticking out of the lock on the outside.
When Rasputin first meets the Empress, he simply walks up to her and starts talking, without bowing or making any other obeisance. This would have been considered shocking and very rude, but the Empress acts as if nothing is wrong or odd.