While the name of the city "Wyatka", the old name of Kirov, is historically correct, the Russian spelling on the station entry is incorrect.
Rasputin's home town was Pokrovskoye, not Petrovskoye, as said in the movie.
Rasputin's beard was much longer in real life than the one of actor Alexander Conte.
At one moment the December Uprising of Moscow "last year" are mentioned, which happened in 1905. However the illness of Czarevitch Alexej Romanow was in 1907.
During the orgy the Russian song "Bublitchki" is played, which was written in 1926, 10 years after Rasputin's death.
When seeing Rasputin walking through the village all people are crossing themselves and touch the left shoulder first and then the right. This is very unlikely to be done in the Russian Empire an orthodox country. The orthodox Christians touch the right shoulder first and then the left when crossing.