Early in the movie when Rusty is at older Russ' house, as Russ tries to sleep he comes out because Rusty is screaming. Rusty says the face on the moon can be seen clearly that night and that is why he is screaming. But in reality the face on the moon is on the side of the moon that faces away from earth so it can never be visible from earth.
When Russ is talking to his assistant about seeing a kid on his property, then goes to sleep he is wearing a blue t-shirt with UCLA on the top left and dark sweatpants, later when he gets up with the bat to check what made the noise outside, he's wearing a grey t-shirt and grey sweatpants.
Russ startles Rusty; Rusty throws his popcorn bowl and the popcorn scatters. When Russ spins the chair, the popcorn is not there, but when we see Rusty again, he and the chair are again covered with popcorn.
The adult Russ Duritz appears to have brown eyes, while the child Rusty Duritz has blue eyes. This could be due to the lighting in the film. (Both actors playing the Duritz character at different ages have naturally blue eyes in real life, so this is technically not a continuity error).
As Russ and Rusty are in the diner for the last time drinking chocolate shakes, they toast. When the glasses clink together you can see that Russ drops his drink but in a split second he has a full cup in his hand.
Russ and Rusty are supposed to be the same person however Bruce Willis (Russ) is left handed while Spencer Breslin (Rusty) is right handed. This is especially noticeable during the scene where Russ and Rusty are playing cards out on the balcony but can also be seen when the characters are are doing other things such as eating.