Serving as an interesting counterpoint to "The Return of Archons", which uses a computer to perpetuate peace, "Armageddon" uses computers to wage war, a creative science fiction idea that was ahead of its time in the 1960s. (It is, of course, also a fine metaphor for the Cold War, with battles happening at a safe distance in satellite countries and draft lotteries to determine who gets to join in the fighting... although the Vietnam draft was still a couple years away when the episode was made, showing how Star Trek was ahead of the curve).
Kirk, of course, will have none of it, believing war should be gritty and dirty and not so nice and neat for a civilization. And besides, the Enterprise has been declared a casualty, setting up a conflict between him and the planet. Guest star David Opatoshu, serving as the planet's spokesman, Anan 7, is perfect as the mealy mouth leader who enjoys giving bad news with a smile. As he and Kirk debate, it's fun to see him lead the captain in circles until Kirk has had enough and decides to just start blowing things up, which is kind of a novelty on a planet more used to bureaucratic warfare than the real thing. It certainly doesn't endear him to Anon's daughter, weakly played by Barbara Babcock. (In fact, Shatner never does get to first base with Babcock despite her participation in seven original series episodes. To be fair, for most of her episodes she simply does a voice, like in "Squire of Gothos" where she's Trelane's mother. For some reason, as a physical actor she's rather wooden and not as interesting).
Meanwhile, Scotty is left to deal with Star Trek's most stubborn, obstinate figurehead of the original series (and that's saying something). Ambassador Fox. Played by Gene Lyons (who died just a few years later), Fox is a thorn in everyone's side, but when Scotty refuses to follow his (dangerous) orders, the engineer really "puts the haggis in the fire". It's a nice piece of drama we can all relate to, because we've all had to deal with authority figures who make bad decisions, when they make unsafe demands, it's important to put a foot down and simply say no. (James Doohan, himself, went through this as a lieutenant in the Royal Canadian Artillery when he refused a Colonel's order to set up a dangerous training exercise).
With the A and B stories working hand in hand, the episode moves along swiftly (even if it does go in circles, thanks to Anan and Fox).
Remastered: This a pretty basic effects episode, borrowing footage of the Enterprise in orbit from "The Naked Time". For the upgraded version, a more realistic Earth-like planet is featured, and some fancy new shots of the Enterprise are included. (They even show the Enterprise in orbit of the planet's moon for one shot to cover a script problem. Somehow, Ambassador Fox beams down to the planet, despite the Enterprise needing her screens up for protection and "Arena" establishing the common sense idea that no one can beam through the ship's screens when they're up. By having the beam-down occur with the Enterprise behind the moon, CBS makes it look like Scotty is ingeniously using the moon as a shield, allowing him to temporarily drop the screens and carry out the action before raising them again).
The original version is probably most notable for some live action footage integrated into another gorgeous Albert Whitlock matte painting, representing the planet's main city. (They actually reuse a studio set from "The Menagerie", which is why Kirk appears to be standing in front of the same wall as that episode, though the matte extension is different). Sadly, this is the last integration of artwork and live action in the series; though happily, CBS Digital makes the most of it. They enhance the matte painting with more realistic features and the hustle and bustle you'd expect from a populated area (including a light-rail system), and they add a second shot to go along with, showing more of the city. None of it is particularly spectacular, but it's all fitting for the kind of episode it is.