The lighting in the park bench scene with 'Dr. Juliet Devereau' and 'Max' alternates between flat overcast skylight and bright sunlight.
(at around 19 mins) When Juliet and Max sit down on a bench in a park to talk, Juliet crosses her legs, but a second after her legs are not crossed.
(at around 1h 4 mins) When Juliet calls the hospital to inform that she will be late, she obviously doesn't talk to anybody because we see the locked screen of her iPhone with the typical "slide to unlock" screen. It's not possible to make a call while an iPhone is locked.
During the scene where Max sneaks into Juliet's bed room and tries to kiss her while she's sleeping. When he's leaning in, she is laying on her side, then the camera angle changes and she's on her back, then the camera changes back and she's back on her side and he kisses her. Then she rolls over onto her back and Max stands up.
When Jack is carrying groceries into the kitchen, you can see the cameraman's reflection in the window.
Max tells Juliet that the apartment is $3800 + utilities and states that Con-Ed (electric) will be expensive in the winter, however it is quite clear from the radiators in the apartment that the entire building is heated with a hot water boiler.
When Max is showing Juliet the apartment he says "Well, there's the F line". No New Yorker would ever refer to a subway line that way; he would say "the F TRAIN ".