In the pilot episode, Chandler and Joey's apartment number is #4 while Monica and Rachel's is #5. Later in the season, the numbers change to #19 and #20.
In the pilot while Rachel is on the phone, apologizing to the guy she left at the Altar, [in the extended version] she says his name is Barry Finkel but in later episodes it is changed to Barry Farber.
When Monica is saying bye to Paul, Paul goes to the screen left but the stairs are at the screen right.
At the end of the episode when Ross picks up his jacket, there is a blue bowl on the small table by the pillar and nothing on the main kitchen table. In the next shot, the blue bowl has moved to the large table.
When Paul the Wine Guy first comes to Monica's apartment, Phoebe is standing in the kitchen telling Rachel that she just pulled out four eyelashes. But in the very next shot, you can see her sitting in an armchair in the living room. [Broadcast version only - in the extended DVD version there's a shot showing Phoebe walking to the chair.]
As Monica is walking to her room and steps on Paul's watch, if you look carefully, you can see that she just steps on the band. That would not damage the watch.
When Rachel doesn't think she can cut up her credit cards Ross quotes Rachel by saying "Come on, you made coffee you can do anything". However Ross wasn't there when she said it.
While in Ross's apartment Joey explains that women are like ice cream and he should grab a spoon. Ross implies he hasn't "grabbed a spoon" since the song Billy Don't Be a Hero came out. However the song came out in 1974 making Ross 6 years old at that time.