After he works out the sequence of moves which will win the game, Josh offers Jonathan a draw before making his move. Proper chess etiquette dictates that a player only offer a draw AFTER making a move.
There are numerous mismatched shots of games in the film. In the final game, Josh moves his queen early in the game, yet later it can be seen on its starting square. In the first game he plays in the park against the old man who beat Tal, as he makes his first move there are already several pieces from both sides deployed on the board. In the first game against his dad, the father picks up a bishop in one shot and a second later sets down a rook to mate Josh.
In the opening scene when Josh encounters Vinnie for the first time, as Vinne stands there with the baseball pointing at the chess piece the sunlight is over his left shoulder. In the reverse angle shot of Josh, the sunlight should be over his *right* shoulder, yet the light is on Josh's left.
Josh tells Vinnie he learned the "Schliemann Attack" from his teacher. Actually there's no such thing. It's the Schliemann Defense. Also, when Vinnie asks, "What's that?" referring to the "attack," Josh actually hasn't yet made the move that would prompt him to say this, he's just playing the opening move of the standard Ruy Lopez (1. ... e5), so there'd be no reason for Vinnie to say "What's that?" at that point. The move that makes it "Schliemann" (3. ...f5) hasn't yet been played.
"Errors" such as the McDonalds cups seen in the movie being from the early 90s and not mid-80s are not goofs. Although the real life events happened in the mid-80s, the movie makes no attempt at all to stay in period. The cars, computers, clothes, etc., are contemporary to the film and not from the 80s.