It is shown several times that a key is needed to get into Kathleen's apartment building. However, when she leaves her building to go meet NY152 for the final time, she does not have her keys (and no purse or pockets).
When Joe and Patricia get stuck in the elevator, the elevator man suggests they all jump up to trick the elevator into thinking no one is inside so the doors will open. Yet you know from their entry on the ground floor (and earlier scenes) that the elevator doors are operated BY HAND, and by that same elevator man.
In the scene where Joe is typing his message about "The Godfather", Joe types "Don't lose you're nerve." This is clearly visible on screen both as it is being sent and as it is being read. It should read, "Don't lose your nerve" - "you're" is the contraction for "you are".
When Kathleen is in the cafe waiting for NY152 to show up she rearranges the book and flower and ends up putting the flower inside the book, then Joe comes in and she picks up the book to cover her face and the flower is now on top of it and the book is in a different position.
When Joe comes to visit Kathleen at her apartment when she is sick, he tells her to sit down at the table. When she sits down, there is a large pile of tissues on the table, but in later shots, they are gone.
At a party where Kathleen realizes that Joe is in fact Joe Fox, she is talking to him as he helps himself to the caviar garnish from a plate. He scrapes a section of the garnish from around the edge of the plate. When he does this again a moment later, the caviar garnish is back in the exact same position it was before he first scraped it off.
In the final scene at the 91st St gardens, when Kathleen and Joe kiss, a hot dog vendor behind them opens up his umbrella fully, and it is open for the rest of this kiss. A second later, as the film cuts to the dolly/crane shoot pull-away, the vendor is again opening his umbrella.
When Frank leaves the apartment at the beginning of the movie, the computer is at a table at the end of their bed. After Kathleen checks on him as he leaves the building and runs around to sit down at the computer, it is now on a table facing the living room furniture.
When Kathleen writes the email to Joe about how she just told off someone at the café, Joe reads the email and logs off. You hear the AOL voice say "Goodbye."; so when he replies to her email, he just begins typing the email, without logging back on. AOL allows you to write emails offline.
Toward the end of the movie, Joe and Kathleen are standing at a hot dog stand by a window. To avoid reflections/glare there is no glass in the window, which would be fine except that the glass is present in adjacent panes and the reflections of passers-by come and go as they walk along the street.
When Kathleen is telling Joe he will love Pride and Prejudice and he is trying to read it, it clearly shows the Colin Firth cover. When Joe picks up the book Kathleen has brought at the café and says 'Pride and Prejudice, I bet you just love this book', it is actually the cover of Austen's Persuasion.