Deela: The species is capable of much affection.
Rael: I have noted that.
Deela: I wonder if they will demonstrate it to us.
Deela: We are handling it in the only way we know how. The way our parents did, and their parents before them.
Kirk: Did they solve anything?
Spock: Mr. Scott, we cannot cope with them on our level.
Scotty: Can we find some way of coping with them on their's?
Spock: That is a very logical suggestion.
Deela: Don't you dare do anything like that again. It's contemptible.
Rael: Then don't torment me. You know how I feel.
Deela: I don't care what your feelings. I don't want to know that aspect of it. What I do is necessary and you have no right to question it. Allow me the dignity of liking the man I select.
Deela: Do I displease you so much?
Kirk: I can think of nothing I'd rather do... then stay with you. Except staying alive.
Scotty: Captain Kirk, where the blazes did you come from?
Kirk: Out of the nowhere, into the here.
Kirk: Mr. Spock, my compliments to your repair work and yourself.
Spock: Thank you, Captain. I found it... an accelerating experience.
In the opening sequence, Scotty has a different hairstyle as he dictates his log, and Chekov briefly appears despite the fact that Hadley is at the navigation position the rest of the episode. This is because the sequence uses recycled footage from "The Empath."
For the Scalosians to beam aboard the Enterprise, they would have had to stand still long enough for the transporter to (from their viewpoint) very very slowly materialize them. If they stood still that long they would be seen. Also it makes no sense that they could "secretly" be beamed up. The transporter operator has to know how many people he's beaming up.
As Compton takes the water samples in the opening sequence, you can see him wipes his mouth with his wet hands. There's two problems with that. 1) It's unlikely he swallowed, but he disappears faster than Kirk when the captain later ingests it directly. And 2) there's no way a trained crewman would wipe potentially contaminated water on his mouth, or even let it contact his body.
There is no explanation for the selective forcefield that throws back the security guards. It is never referred to again, nor is there any indication that the Scalosians have or need a selective forcefield.
When Kirk begins to accelerate on the bridge, he turns to see Scotty who is standing up and then freezes motionless. But when Deela appears, Scotty is seated behind her.
When Kirk approaches Spock on the bridge and demands answers, Spock briefly swivels very slowly on his chair. But at this point everyone is frozen motionless: Spock shouldn't be moving at all.
The "frozen" Uhura's earrings move as Kirk walks past her on the bridge to talk to Deela, even though they should be frozen as well.
Even before Kirk drinks the coffee, the turbolift doors behind him are frozen open even though nobody is going in or out of the turbolift.
When Kirk records the message for Spock, the recording unit is is at waist height directly in front of him. But when Spock plays it back the camera angle is considerably higher and several feet away.
Even overlooking the convenient explanation of radiation, the physics of this episode don't make any sense. The Scalosians have to move at least 3,600 time faster than a human to remain invisible. That would be approximately 1,680 miles an hour: Mach 2. There are no sonic booms, and they ignore inertia as far as stopping and turning corners. It's stated the Scalosians are on-board 8 hours, or 480 minutes. 480 Scalosian minutes = .13 human minutes, or about 8 seconds. Clearly more time takes place for the Enterprise crew then that. Even waiting for an automated door to open would take a Scalosian hour (all of the doors are conveniently if inexplicably open).
Kirk fires a phaser at normal speed and Deela dodges. However, the normal-speed Enterprise crew never see it.
Although Scalosian water could be an extremely useful weapon in Federation hands, as long as the user was careful and had the antidote nearby, no mention is ever made of it again.
No one attempts to offer the Scalosians the antidote. Even if there's a risk, it seems they should at least have the option, rather than being quarantined for eternity on Scalos.
When Kirk relieves Scotty on the bridge at the beginning of the episode, you can see the shadow of a stage light just to the left of the monitors above Uhura's head.
What Changed in the Remastered Version
The planet Scalos gets a nice orbital upgrade. A better matte shot over Kirk's shoulder suggests a much more expansive city as the site of the first landing. The phaser effect when Kirk, Spock and two security men tried to penetrate the Scalosian force field got a very nice upgrade, as did Kirk's attempt to stun Deela.
Original vs. Remastered
The idea that radiation could produce a permanent, heritable metabolic shift of this magnitude seems silly, but one must remember that when this episode was penned, the subtler, long term effects of radiation exposure were less well understood.
Early in the episode, Rael explains that cell damage in those newly accelerated causes rapid aging and death. Were this the whole story, no one would survive acceleration, for cells constantly undergo apoptosis, a programmed process where they disintegrate in an orderly way and their resources are reclaimed. Generally, age triggers this effect. Tens of billions of cells disintegrate this way every day. Presumably, then, the rapid aging effect must require significant cell damage (as from a cut) or cell damage from necrosis (an unprogrammed death that costs the organism resources).