In the pilot, the Huxtable living room design is completely different in the rest of the show.
The exterior shots of the Cosby house shows a typical New York style attached town-home. These have solid common brick walls that separate the units. Therefore, there wouldn't be a window in the stair well in the living room showing the outside, the only thing on the other side of that wall would be the neighbor's living room.
Claire asks "Why do we have 4 children?" In the rest of the series, they already have 5 children.
The girls, and Claire refer to Theo as "Teddy" in this episode.
When Cliff is talking with Denise's date, the date comments about being 'in a Turkish prison' but doesn't reveal how long he was in one. Later, upstairs, Cliff warns Denise about dating someone who had spent a year in a Turkish prison. How did he know that information if the date didn't tell him?