Both Beth and Harry Beltik refer to how they've changed since they first encountered each other "five years ago". But the Kentucky tournament took place in October 1963, and the timing of the start of this episode is right after the 1966 Mexico City invitational and before the 1967 US Championship in Ohio, so they first met approximately three years prior, not five.
When Beth is playing speed chess with Watts, after one of the games Watts is holding some money on the table, but as soon as the camera changes angles, the money is gone.
At approx 33:48, when Beth leaves the dorm to go to the student union, she takes a coffee cup that is of normal height that has a brown stripe around the top. She enters with a cup that is whiter, taller and has no stripe.
The mug Beth leaves with is different from the one she arrives with.
At 20:05 Beth sets a dishtowel down on the counter. It changes position in the next shot when she picks up the plates.
When Beth enters her house after returning from Mexico, the phone rings. When she goes to answer it, you can see that it's a push-button phone. They would not be available for another ten years.
In Paris, the tournament was a round robin so there would be no match labeled the "finals" and all of the players would play the last day.
Returning from Mexico (1966) Beth answered the kitchen phone. It had 12 digits. 0-1 and *, #. Push button phones only had 10 digits. 12 digit phones with the added *,#, keys, weren't available until 1968.
When Beth gets into bed and reads Borgov's book she holds it upside down.
In the fourth game of speed chess between Beth and Benny, Beth's flag on the clock has already dropped before she resigns.
Margaret's baby at 18:09 is visibly a mannequin.
In the newsreels of Borgov there is a picture of Brezhnev on the wall. This would have been a picture of Nikolai Podgorny in 1967.
When Beth goes into the drug store, the scene opens with the camera panning a shelf of goods. In between the Little Debbie's marshmallows and corn starch is a number of circa 2020 Kedem grape juice bottles.
When Benny drives Beth to his home in NYC, he parks on the street behind a Ford Mustang. It was a 1968 model, and the episode was set in 1967. (It could have been an early sale of the car, starting in Sept. '67, but the timing of the episode is unclear.)
When Beth answers the phone upon her return from Mexico, the telephone is a push button phone, which did not come out until late 1976.
When Harry is handing over books, he mentions one by "Alekhine", pronounced as written. Correct pronunciation is "Alyekin".
At around 35:36, when Benny invites (a reluctant) Beth to play speed chess, he provides a list of other chess-related games to play instead, such as skittles, blitz, or bughouse.
Blitz and speed-chess are exactly the same, and a player like Benny Watts would know that.