Back in 1980s the only foreign public bus purchased by USSR was Hungarian Ikarus, while definitely different buses were used for filming Moscow. Also the truck selling gasoline suppose to be military judging by its khaki color, which MUST be Soviet made, but is obviously foreign.
In Ivanov's Moscow apartment he greets his family "utro" ("morning" in English). But it doesn't work in Russian. The only correct way is to say it in full "dobroe utro" (good morning). Obviously but mistakenly copied from common way to cut it short in English - "morning".
In his apartment Vlad plays a tape for Lucy that he says he brought from Russia. He defected in Bloomingdale's with only the clothes on his back so it is doubtful that he had a cassette tape with him.
Right at the start of film you can see crew reflected in window as Robin Williams is headed to the line to get shoes(right when the Soviet Officer catches up to him as well)
The can of Coke that the KGB agent is holding during the circus performance in New York is obviously not an American one (the sides go straight into the rim, rather than a slight lip between them).
Orlando and Vladimir's "Korean Cab Driver" speaks in Japanese after Orlando berates him for missing a turn. ("Gaijin urusai! Monku, monku!" - "Annoying foreigner! Complain! Complain!")
When Antoly writes "FREEDOM" on the bus window, some incorrectly believed he was writing it in dust on a previously clean window. He was writing it in condensation.
Whenever Vladimir plays the saxophone, he is clearly not playing the music heard (the fingering is off and there doesn't seem to be any particular rhyme or reason to his breathing).
Road signs in Moscow scenes are messy. For instance, Vladimir passes sign "no vehicles allowed" and some meters after you can see the sign "parking forbidden" which means a vehicle suppose to disrespect 1st sign to succeed in disrespecting then the 2nd.