When Castle is talking to Beckett while they are in bed after finding Cole's body, the pillowcase under Beckett's head is gray and patterned. When she wakes up in the morning, it is a solid beige pillowcase.
Castle had his binoculars focused to see Cole at his car, a distance away. When Cole popped up right next to Castle's car, Cole was still in focus. He should have been blurry through the binoculars.
When Castle is spying on Cole, Cole notices him so Castle tries to hide. When he sits back up, there is a shot of Cole's car that it black around the edges as though it is still through the binoculars, but Castle isn't looking through them. He brings them back up to his eyes later in the scene.