When Data is playing craps, Riker says "the probability of rolling a six is no greater than rolling a seven" implying they have the same odds. This is not true. The probability of rolling a 7 is higher (6/36) than the probability of rolling a 6 (5/36).
Riker notes that since the flag patch on Col Richey's uniform has 52 stars, it "places it between 2033, and 2079 A.D." However, when requesting information about the colonel, he states that they are in a 20th century structure, and states that Col. Richey is from the same era. Those dates are 21st Century, not 20th.
When 1st playing blackjack, the dealer's card changes from a Q clubs to a K diamonds. When Data comes back to ask questions after being in the hotel room, the dealer's card is also a K diamonds. Obviously the shots somehow overlapped resulting in the 1st BJ scene's cards changes.
When the revolving door is viewed from the outside, Worf enters first, Data second; then Riker misses the third section and enters using the fourth section of the door. When the revolving door is viewed from the inside of the casino, Worf enters first, Data second, and Riker enters immediately after Data, rather than using the fourth section of the door.
When the away team finds the body of the astronaut, they make reference to 52 stars on the American flag patch. Yet, when the piece of space debris is displayed by Riker in the beginning of the episode, the American flag below the NASA logo only has 50 stars.
Early in the episode Geordi states that the surface temperature of the planet is -291 deg C. This is physically impossible as the minimum temperature possible in the universe is 0 deg K (-273.15 deg C)
After beaming the ship debris aboard, Riker immediately picks it up. But the metal would still be as cold as space (-270 Celcius [-454 Fahrenheit]) which would freeze the skin off of his bare fingers but he shows no ill effects.
Riker states that the 52 star flag patch "places it between 2033, and 2079 A.D." A.D. stands for "anno Domini," which translates as "the year of our Lord." A.D. is properly placed in front of the year number, not behind.
When Commander Riker leaves the elevator in the Hotel Royale by himself, the door quickly opens and immediately closes after he steps out. Since the story of The Hotel Royale takes place in the 20th century, there are no hotel elevators whose doors would open and close so quickly.
The goof item below may give away important plot points.
When the away team first speak to the desk clerk, he tells them "We've been expecting you, a trio of foreign gentlemen." However, at the conclusion of Data's winning streak after Riker offers to buy the establishment, the clerk seems surprised to learn that the away team are the foreign investors spoken of in the novel. Since there was no mention of a 2nd group of foreign gentlemen, it seems only logical that the same foreigners the clerk encountered at the start of the episode would be the same ones buying the hotel, thus he shouldn't be surprised.