When Joel is pacing outside Molly's shop, and she is trying on outfits in the back, the shop is neat and tidy, and the Kex poster on the wall is straight. When she finally comes out, in a suit of armour, the shop is trashed and the Kex poster is askew. This is explained in the bonus material deleted scenes, which shows two CSR employees trashing the shop with baseball bats.
After Joel "finds" a cheeseburger behind Tucker's ear and gives it to him, Tucker hugs him and goes upstairs without the burger in his hands.
When Joel and Molly are seated in the big picture window of a coffee shop, gabbing, the sign for the subway stop across the street, although partially obscured, clearly says "__ Street Subway Station" (around the 31-minute mark). A minute later, the same sign says, "Upper W. Side Subway Station". Neither sign really exists along the 2/3 line.
(at around 1h 2 mins) When Joel is watching slides, the clock on the wall behind him indicates 01:50.
Later, he is watching Joel and Molly's hologram, the clock still shows 01:50.
When Joel first meets Tucker (Mollly's son), Joel says: "[Smeargle] can defend Butterfree's Hyper-Reverse attack with two double-colourless energy cards, and only get 10 damage points". This is, in fact, not at all how the Pokemon card game works. Double-colourless energy cards were banned long ago, and are only used in legacy tournaments. Even in a legacy tournament, Smeargle would not be permitted as a legal card, as it appeared after double-colourless energy cards were banned (Generation II).
After bumping into each other, they both leave their groceries on the sidewalk.