I feel guilty to mention this after not finding most of the mistakes in "Lord of the Rings", but I couldn't miss two mistakes- both anachronisms- in "A Beautiful Mind."
First, they show Mrs. Nash in a scene in the '50s using a Tupperware Servoliere container (the squarish ones with the pleated-looking tops). If I'm correct, those didn't come out until the '60s or '70s.
Second, in the 1998 scene where Nash has tea and receives pens from his colleagues, they have a container on the table with packets of sugar, Equal, Sweet n' Low, and Splenda. This last product wasn't available in the U.S., especially in packets, until at least 2000.