Incorrectly regarded as a goof: Boimler is able to sneak into the Captain's Party by posing as his transporter clone, William, who serves aboard the USS Titan. He is allowed in because the Titan is a Lunar-class ship, while Mariner is rejected because her record accurately reflects her service aboard the USS Cerritos, a California-class ship. Neither one are command level officers, so neither should have been allowed entry based on that criteria alone, regardless of what class of ship they served aboard.
However, the after-party follows the "Command Conference" which is assumed to be for command level officers only. There is no mention of the criteria to enter the after-party, however other than the attendee must be assigned to a "Luna-class and above only" according to the bouncer. Lieutenants Junior Grade are obviously allowed since Boimler's clone is on the guest list.
Incorrectly regarded as a goof: Boimler and Mariner refer to the dress uniforms as "skants." A skant was a type of duty uniform seen early on in the run of Star Trek: The Next Generation (1987). It is different from a dress uniform.
The two uniforms Boimler is holding are much longer than a dress uniform.They come down to about his knees, even longer than the skants seen on TNG. They are definitely supposed to be skants.
It is revealed that, back in the 2260s, Kirk and Spock drank at the same bar as Boimler and Mariner after failing to successfully crash the command party. Kirk and Spock served aboard the Enterprise. At the time of their service, the Constitution-class was the most advanced starship type in the fleet (later Star Trek shows have retroactively designated the original Enterprise as the flagship, a title it never held during the original run of the series or its sequel films). Boimler and Mariner were denied access to the command party because service aboard a California-class ship is not considered prestigious enough to warrant entry, but service aboard a Constitution-class would not have carried that stigma. Kirk and Spock would not have had to crash the command party.
The crew of the Cerritos is able to reverse the Doopler ambassador's duplicating by angering him and his duplicates, which triggers a recombining, something which is treated as though it is a new discovery. Surely the Doopler would be familiar enough with his own species' anatomy that he would know about this phenomenon.
When Boimler finds Mariner in the dive bar, he orders a Prune Juice Spritz. However, the barmaid presents both of them with the same drink.
It is revealed that, prior to losing part of his memory, Rutherford left notes on the components of the Cerritos model to slow down his future self's progress in completing it. That would only make sense if he'd known he'd be losing his memory. Had he not lost his memory, he'd have known that what the notes told him wouldn't work because he'd have remembered writing them.
After the crew is finished dining with the Doopler ambassador, they start to discuss the upcoming after party. Dr. T'ana and Lt. Shax are shown facing each other. Dr. T'ana's right ear is shown as perfectly straight with no injury, instead of the normal missing bite damage.