When Laura Gittleson, the real estate agent, is killed by Dr Freudstein, we see her body lying on the floor with her eyes closed and her mouth open. A few moments later, Freudstein drags her into the basement and her eyes are now open, but her mouth is closed. When she is hanging in the basement near the end of the film, her eyes and mouth are both open.
Norman's body changes positions in the cellar.
Fishing line can be seen attached to the bat that attacks Lucy in the cellar. Also, when Norman stabs the bat with a pair of scissors, blood comes out of the bat in places other than where the scissors penetrated it.
When the real estate agent is stabbed in the neck by Dr. Freudstein, he pulls the poker away. For a split second, you can see that there is no actual wound where the woman is stabbed.
When Dr. Boyle and his family arrive in Boston, you can see the crew's reflection on the car before Mrs. Boyle opens the door.
The family is leaving New York and moving to Massachusetts. They are shown driving southeast, over the Brooklyn Bridge, away from Manhattan (the former Twin Towers are in the background). This would put them on Long Island, which is an island. There is no way to get to Massachusetts from Long Island.