Plot hole: The very ending scene where Peter (in this version died in the ice as a boy so Walter's wife never met him) takes off some of the dust covers then picks up a toy airplane. The toy airplane he got as a child from ... Walter's wife when she was trying to placate him in an earlier episode for trying to run "home". This toy airplane wouldn't have been his as a child in this version of the universe.
Factual error: When discussing whether Dr. Nayak's server could have been hacked, Agent Broyles says there was "no sign of a DDoS attack." A DDoS (Distributed Denial of Service) attack is not a means of hacking into a server to alter data - it refers to flooding a web site with so many connections, through automated means, that legitimate visitors can't connect to it. It's a mode of attack that has no relevance to the situation they were discussing.
Revealing mistake: When the typewriter is being used at the end, the keys being pressed don't correspond to the words appearing on the paper. Obvious from the start - the first letter pressed is on the middle row of the keyboard, but the first word typed is "unacceptable".
Continuity mistake: At the very end of the episode when Nina Sharp exits the elevator, she mostly steps out of the elevator completely before the masked men come and shoot her. However, when they review the security footage of the attack in the next episode, "There's More than One of Everything," you see that she barely takes a step before being shot.