Star Trek was quite different as a show from any of the later incarnations. It was an action/adventure show, for the most part. Even when they were being silly (Piece of the Action, for example), there were usually elements of adventure involved. The other shows are not adventure shows (except when the Borg show up, in which case all normal episode mechanics go out the window, which is why the Borg episodes are generally considered to be the most like the shows from the original series, and often considered fan favorites).
Star Trek also had a chemistry among its lead characters that none of the other shows ever managed to replicate. What made a show like MAS*H fun to watch wasn’t so much the jokes (telling jokes is easy) as the existence of a relationship among the characters that allowed for interesting dialogue and plot development.
A good example of this is the episode “This Side of Paradise.” As an audience, we find ourselves torn during the episode. On the one hand, to see Spock able to be liberated from the confines of his self-imposed emotional control makes us happy for him: he can LOVE! On the other hand, we worry about the ship and the crew, and feel a sense of angst over the fact that the story appears to be leading to a breakup of our familiar space-faring family. How can we be happy about a Spock who won’t be at the computer, helping the Captain in the crucial moment with some vital bit of data? And the struggle the Captain goes through pulls us in as well. We identify Kirk with the Enterprise, as if they are a unit, more than a marriage. How painful must it be to yield this for happiness?
One of the all-time best sets of scenes in the show comes in the second half of that episode. Yes, Shatner can’t act his way out of a paper bag (I Can’t LEAVE!), and yes, the situation is a bit contrived, and yes, all the usual plot holes exist. But from the moment Kirk comes to his senses in the transporter room, through his harranging of Spock (“right next to the dog faced boy!”), to the tearful scene where Spock turns his back on happiness, and admits to Leila that it’s a self-imposed rejection of her love, we cannot help but identify with, and feel for the characters. We want a resolution, yet the resolution, when it comes, leaves us grieving.
There were sooooooooooooooooooooooo few situations like that in the later series, and it’s in large part due to the fact we never really identified with the characters and cared about their interaction, which always seemed a bit forced. In the few episodes where they allowed character development, it drew us in (“The Inner Light” for example) and we cared. But this sort of caring was much more prevalent with the original series.
I’ll always be a bigger fan of the original series, despite the fact that it was far less “good” in certain classical senses. I always wished they would dump their stupid A-plot, B-plot silliness from TNG, and get back to some real ACTION episodes, with characters who you could really get into and believe in. When Tasha Yar was killed off, I remember thinking, “Eh.” When Wesley Crusher left, I though, “Good riddance.” I’d have never thought those things about anyone in the original series, even Riley.