When Archie hands the baby to his son, Ezra, there is baby vomit on the back of Archie's left shoulder. It disappears in the next shot, but shows up again when Archie is standing there watching the baby leave.
After Archie leaves his house for Vegas you are shown a aerial view of Brooklyn. If you watch the bridge in the lower right corner you can not only see the shadow of the helicopter taking the footage , but the helicopter footage is rewound as seen by the helicopter floating backward.
After discovering that Binion's had no rooms, they headed to Aria. in the next scene, they are walking with their luggage on the strip heading north up the strip away from Caesars palace. Aria is in the other direction, south of Caesars Palace.
When the group gets out of the taxi after getting suck in traffic on the Las Vegas strip in order to walk to Aria, they get out in front of The Mirage and begin walking north, however, Aria is south of The Mirage.
When Archie buys in at the blackjack table for $15,000, the pit boss would have stopped the transaction and asked him for ID and his social security number. Any cash transaction of over $10,000 requires this.
Just a little after the start of the movie, Billy gets Sam and Archie on a conference call. Since he is the one who creates the conference call, when he disconnects, automatically all connections should be broken. However, in the movie, Sam and Archie keep talking even after Billy hangs up.