Adaptation Displacement: The movie is much more well-known than the play, even though the opening credits actually state in big letters what it was based on.
Genius Bonus: Many historical events mentioned in the movie are real. The song sung by minstrels before Bunsha interrupts them is an actual old Russian song about the battle of Molodi. The Crimean khan's troops marauding near Moscow is an actual event from that same war (though they mixed up the order in which those events happened). The cruel and unusual execution by barrel of gunpowder also happened, with the exact same phrase uttered by Ivan IV, though the victim was a seditious aristocrat, not an inventor.
Older Than They Think: The Kremlin shown in the film is made of white stone, which is its original look. However, it was rebuilt to its current "red brick" look by Ivan the Terrible's grandfather Ivan III.
Ironically, before the Red October is was customary to whitewash its brick walls every couple of years to imitate the original white stone look. Even when the Soviet Government put its seat there, they infrequently continued this practice, allowing the whitewash to fade to the original brick color in the intervals, and painted the walls with a fake houses and windows during the World War II as a defensive measure. The modern practice of painting the Kremlin walls red actually started only in 1947, in preparation of Moscow's 800-year anniversary.