When Pete and Beth are sitting on the bed in the hotel room, Beth takes Pete's left hand, kisses it and holds it in her lap while she is talking to him. When the camera switches angles, she is no longer holding his hand.
When Don meets Peggy in a cinema, they stop talking as the lights go down and we hear the opening of the film - Burt Bacharach's main title music from the 1967 version of "Casino Royale". However, this film has a pre-credits sequence first, so we should have heard dialogue instead.
Near the end of the episode in a motel, Peggy appears to have come out of the shower - hair wet and in a robe. She looks outside the window then sits on the bed and stretches out her legs. The bottom of her feet are too dirty for her to have had a shower.
The "Earthrise" photograph visible on the wall in Pete's office was shot during the Apollo 8 mission, which did not take place until December 1968.