Continuity mistake: After Spike finishes reading his poem to his mother, the piece of paper that it's written on disappears from his hands between shots. We never hear or see him put it in his pocket and he doesn't move to put it down on the table beside him.
Revealing mistake: When Spike tells his mother that he's become a vampire, she asks him if he's drunk. He responds with "Little bit." Just after he says this, someone can be heard coughing off-camera. It could be his mother because she isn't supposed to be well, but it seems odd that the camera wouldn't be on her during this. And a cough wouldn't really be necessary for the scene.
Other mistake: Robin uses the song that Spike's mother sang to him to provoke him into attacking him. The song is played on his computer through Apple's media program "iTunes". He clicks his computer's mouse to start the song, but when we see the monitor the cursor isn't placed on the Play button. Robin doesn't jostle the mouse when he presses it, it remains completely still on the desk. And the only way to play a song through iTunes using a single mouse-click is by clicking on the Play button.