When driving to Times Square, Ray remarks on how nice it was "after urban renewal". The urban renewal didn't even start until Rudy Giuliani became mayor in 1994, three years after the game takes place. (PC/PS3/X360 version only.)
Whenever Egon introduces a new pack upgrade, the additional hardware just suddenly appears on the pack. It's particularly noticeable with the introduction of the slime blower, with a green tube running along the cable from the pack to the hand unit that wasn't there mere seconds before. (PC/PS3/Xbox 360 version)
In the library, when you first see "the Grey lady", Ray says "That's her! Elenor Twitty - The Grey lady!", seeming to already know what her name was when she was alive. But later, when the Ghostbusters are in the basement, Winston reports that a librarian named Elenor Twitty was killed in the 1920s. Ray says "The Grey lady?", to which Egon replies, "That would be my guess.", implying that this is the first time the Ghostbusters have heard the Grey lady's real name.
The so-called "Grey lady" has been bright purple every time she has been seen. Why would she be dubbed "the Grey lady"? Because "grey lady" is a common paranormalist term for an elderly female ghost, regardless of the coloration of the apparition.
When listening to messages on the answering machine back in the firehouse, if only one message remains on the machine it will say "You have one new messages." However, this is not so surprising, especially considering that it is 1991 technology. In those days, the extra circuitry required to eliminate that minor grammatical error in an answering machine may have substantially increased the unit's cost, and so been deemed unnecessary by the manufacturer.