The lash marks on both Kirk's and Spock's bodies disappear and reappear between scenes.
When Kirk and Spock are in the prison and getting flogged, Isak, observing this, has blood dripping from his mouth which changes from the right side of his mouth to the left.
The first Ekosian to confront Kirk and Spock is wearing the rank insignia (3 pips) of an SS Untersturmfuehrer (2nd Lt). After he is overpowered and Spock emerges in his uniform the rank insignia now has the 3 pips and stripe of an SS Obersturmfuehrer (1st Lt).
The second Ekosian to confront Kirk and Spock is wearing the rank insignia (3 pips) of an SS Untersturmfuehrer (2nd Lt). After he is overpowered and Kirk emerges in his uniform, the rank insignia now has 3 pips (in a straight line instead of slanting down) and a stripe, which was not an actual SS rank.
When Kirk and Spock penetrate the Nazi headquarters to attempt to see John Gill, Kirk notes that "They've got the booth curtained off." When he says that, there is a camera shot of the curtain with no one near it. However, prior to that and just after that, Nazis are seen standing right in front of the curtain. There was not enough time between shots to account for their disappearance and reappearance from in front of the curtain, and the action continues clearly indicating that there is no time lapse between shots.
Gilbert Green is credited as "SS Major" but is wearing the Oak Leaf of a Schutz-Staffel Standartenfuhrer (Colonel), not a Major.
In the prison cell, Spock holds the crystals at a right angle to the eventual direction of the beam, rather than aligned with the direction of the beam.
The beam would have come out sideways rather than in the direction shown.
When John Gill is making his speech, the picture is coming from the front (with the microphone, of course, covering his mouth, so nobody can see he's not really the one speaking). However, when Kirk breaks into the room later, the camera is on an angle to his left, so his entire face should have been shown.
It appears that the Ekosian rank structure does not match exactly with the SS ranks of Nazi Germany. The senior officer that Kirk and Spock first encounter is addressed as "Major" (even by another Ekosian), but he wears the rank patch of a Nazi SS Colonel.
When John Gill is making his speech, the picture is coming from the front (with the microphone, of course, covering his mouth, so nobody can see he's not really the one speaking). However, when Kirk breaks into the room later, you can see that the camera is on an angle to his right, so Gill's entire face should have been seen from the same angle rather than straight-on.
Several uniforms, such as Kirk and McCoy's, show cuff bands reading "Adolf Hitler". Adolf Hitler was from Earth and never existed on Ekos. However, it is common for an imitative culture to appropriate a symbol that has nothing to do with them, which also fits in with Gill's plan of perfect mimicry.
Contrary to many movies and TV shows, the Gestapo never wore black uniforms. These were worn at times by the Waffen SS. The Gestapo went in plain clothes. However, as this is Ekos and not Earth, Ekosian Gestapo could potentially wear black uniforms if they do not copy every tiny detail of Earth's Nazis.
The first time Kirk and Spock infiltrate the Nazis, they are speaking English and relying on the Universal Translator to turn their speech into Ekosian, a language with presumably no correlation to English. Their mouth movements should appear to the locals as being badly out of sync with the words being heard, like a badly dubbed foreign movie, which would lead to their discovery long before it actually happens. This is normal for The Original Series, and the fact that all aliens encountered by the crew seem to speak English is never really dealt with. The Universal Translator is only mentioned in a few episodes (but not this one). However, in Metamorphosis (1967) and Bread and Circuses (1968), Kirk does point out with a note of curiosity that the inhabitants of certain planets speak English.
While in the jail cell and attempting to remove the transponders from their arms with bed spring, the handcuffs that Kirk is wearing are clearly unlocked and open.
The corridor leading to the Schutz-Staffel laboratory and the corridor leading to the Fuhrer's broadcast booth are the same, with only minor modifications.
When Kirk and Spock are getting lashed in the prison cell, they already have some marks from the whip. However, they are struck repeatedly over the next few seconds, and no new marks appear.
(at around 6 mins) Kirk and Spock begin watching a newsreel that shows what is happening on the planet. Part of what they are watching are real news clips from Nazi Germany, 323-335 years before "Patterns of Force" takes place. A few times, Adolf Hitler can be seen in the footage, but then the news anchor salutes a picture of their "Fuhrer", John Gill, who looks nothing like Hitler.
It is never explained why John Gill had to introduce actual Nazi symbols (the swastika, specific uniforms, the word Fuehrer) in order to have the society emulate certain Nazi principles of efficiency. Also, although it makes sense that someone like Gill could introduce certain technological advances that would take root and blossom very quickly, it makes no sense that the planet looks exactly like 1930s-40s Germany. Why would Gill waste his time introducing old specs for things such as Tommy guns and automobiles? (We know it's because the studio just happened to have those props handy, but the characters don't know that.)
When Kirk and his party discover John Gill in the broadcasting booth, they could've just slipped into a closet or other secluded area and had him and everybody else beamed aboard the Enterprise, since by then they were in communication with it,
Spock refers to Nazi Germany as a "tiny country" which rose to dominate Europe. In fact, Germany was already the largest country in Western Europe in 1933 when the Nazis took over.