The Executive Mansion, as it was known then, is shown as painted white. It was not painted white until after the War of 1812 when the British burned the building. To cover the scorch marks, it was decided that the building be painted white and from that point it became known as "The White House."
When paintings are discovered of John Adams and Abigail, it is said that they should be placed in the "White House". "White House" was a term coined during the Theodore Roosevelt presidency nearly 90 years later. Prior to that it was referred to as the "Executive Mansion". It is possible that the producers purposely did this for recognition purposes. It is unlikely that Mr. McCollough (historical writer of the book the movie was based on) wouldn't have known this.
When John Adams arrived at the White House, he arrived alone in the late evening on November 1, 1800, and at the south front of the house, which did not yet have the famous semi-circular portico, but which had the only functioning staircase. His famous blessing, which endures on the mantle of the State Dining Room, was written in a letter to Abigail, who was still in Massachusetts and arrived some weeks later. The episode also shows the Adamses occupying the East Room as a parlor/office/dining room. This was not the case; the only use of the East Room at the time of Adams' short occupation, was as a drying room for laundry. The present Green Room was used as a dining room, the Blue Room was the entry, and the Red Room was used as a combination office/parlor. The Adams family used these three rooms almost exclusively, with the exception of the one bedroom on the second floor which was accessed via the servants staircase, due to the fact that the grand staircase was not yet completed. Aside from these errors, the depiction of the White House as it stood in 1800-1801 is accurate and well represented.