In the pilot, Clair asks, "Cliff, why do we have four children?" He replies, "Because we didn't want to have five." A few episodes later, Sondra was added to the cast as their their eldest-of-five, who was away at college.
In the first season, Cliff's name was Clifford. In later seasons, his name was Heathcliff. In the first episode of the final season, Clair calls him Heathclifford.
In the pilot, the dining room banister to the upstairs is very different than it is during the rest of the season.
When Cliff and Clair leave Theo alone in the house while they go on a ski trip, Cliff is carrying some skis out. He drops a ski and it knocks everything off Clair's desk. In the next scene, the objects are on the desk again.
The length of Denise's hair between episodes.
Cliff Huxtable is an OB/GYN in New York City who is successful enough to have his practice in the basement of his home. Yet he has a single examination room, no office staff, and (apparently) no waiting area.
It's established that Clair Huxtable was born in 1946 and that Sondra Huxtable was born in 1964 and Denise Huxtable was born in 1968. That means Clair was 18 when Sondra was born and 22 when Denise was. Yet it's also been established that Clair went away to Hillman College and married Cliff after her graduation while she was in law school. It would have been impossible for Clair to do those two things if Sondra was born in 1964 and Denise was born in 1968.