The Next Generation has hands-down the most tangible connections to TOS, which makes sense given that it was closest in RL time. Gene Roddenberry was involved, many of the same writers wrote, and McCoy, Scotty, and the computer were played by the actually-present original actors (and several TOS guest actors play new parts, including Dr. Pulaski).
Also, TNG is the only show that really duplicates the original Enterprise mission: seek out new life and new civilizations. Some of the later shows were more like this than others, but TNG was the last time that pretty much any story from it could conceivably have been on TOS and vice-versa.
From almost the beginning, though, TNG's adventures had a different feel. While Kirk's ship spent most of its time out on the frontier, Picard's spent a lot in relatively known space, filling in the gaps in the outlines the first explorers had made. The cruise ship look to the place (as opposed to TOS's more spartan, functional look) underlined this; Picard's crew might be routinely in danger, but they were rarely uncomfortable.
That crew was, increasingly, another difference. The seven main characters (more like 8-9 at any one time if you rotate between Tasha, Wesley, O'Brien, Guinan, Ro, and Barclay) became a more balanced ensemble than the six or so TOS crew; once you get past Kirk & Spock the number of episodes where another TOS character is really at the forefront rather than serving as an adjunct drops precipitously. This became not just a matter of actor time, but a whole style of working problems whereby Captain Picard for the most part let people decide how to handle their areas of expertise, as long as they weren't overstepping their bounds, slacking off, or breaking Federation principles. As opposed to Kirk, who respected his crew's technical abilities but was generally up close to the problem to exercise his own judgment of the details.
As TNG entered its later seasons, though, it drifted toward TOS in some ways and away in others. The TV dynamic of "breakout characters" dictated more time spent with Picard & Data at the expense of others, and although their relationship is not Kirk & Spock-like their collective story function became inescapably similar.
At the same time, however, the incipient signs of serialization were appearing (about the same time Twin Peaks came on the air). and although TNG didn't really do story arcs longer than three episodes, callbacks, which were rare & celebrated on TOS, became so commonplace on TNG that it might be hard to find two episodes in a row that didn't refer back in some way. However, this doesn't distinguish TNG on a strictly relative basis because all the other series were even more serialized than it was (same could go for surface improvements such as makeup, special effects, and the appearance of women in senior jobs).
Summing up TNG...
Like TOS: creators, missions, eventually Captain/Science Officer team.
Unlike TOS: decentralized ship operations, decentralized storytelling
Deep Space Nine, in terms of the human element, is an almost total break from TOS. All that's left is Majel Barrett as the computer voice in terms of a TOS person doing their TOS job. Plus, the very un-TOS serialization reaches its height, and the show increasingly commits the un-Roddenberry cardinal sin of questioning whether human progress would always be for the good.
Related to the last, but more subtle (though inescapable once you notice it), is how much less human-centered DS9 is in general. Even in the first season, before it had completely distinguished itself from TNG, it was already routine for a DS9 episode to have scenes with aliens talking to other aliens and no humans present at all. And while the Federation is on balance the most benevolent of the various interstellar governments, that "on balance" sums up a lot of probing and soul-searching that are very different from TNG's confidence in moral rightness and TOS's generally unthinking assumption of it (Spock aside).
And then, of course, you have the mission. The Niners do manage to find some life & civilizations, but they're not really seeking them out. Their job is to rebuild a devastated world and forge an interstellar community out of the many forces interested in the wormhole. As a result, DS9 probably has the fewest TOS-compatible episode plots of any of the Trek shows.
On the other hand, though, there are substantial ways that DS9 hearkened over TNG's head back to TOS. At the top, you have a captain with a very performative way of expressing himself. Now, Patrick Stewart may not talk exactly like you and me, but Shatner & Brooks talk in a way that only makes sense as a show they're putting on for those around them. They have big ideas, and big emotions, and they revel in expressing these. Picard had them too, but when he expressed them it was against all his attempts to avoid it.
And, once again, that Captain is best friends with his science officer. Sisko & Dax have less surface sparring than Kirk & Spock did, but the sense of being supremely trusted confidants is very similar. They're not partners in crime in quite the same way, because Sisko tends to delegate nearly as much as Picard did, and therefore Dax's job often has her working with others on or below her own level rather than him. But their personal relationship is at least as close.
Which is more remarkable because DS9 introduces long-term, relatively stable romantic relationships, something completely absent from TOS and which TNG only had in the most abstract way. O'Brien & Keiko, Sisko & Yates, the Daxes & Worf, Odo & Kira all dealt with the joys & trevails of love with an impact very different from the doomed one-episode-stands of TNG & TOS.
Another thing DS9 brings back is the frontier vibe that was a foundation of TOS. As one of the creators said, if TOS was Wagon Train in space, DS9 was Deadwood, complete with gruff sheriff, wily barkeep, and fiery native woman. DS9 becomes home to the Niners, but unlike Picard's Enterprise it's never really "like home". It's the Niners themselves who adapt.
DS9 is often considered "un-Trek", but I think those similarities are substantial enough that when it paid tribute to TOS, whether through Tribbles or mirrors, it felt like they really knew where Trek had come from, as well as the new directions where they wanted to take it.
Summing up DS9...
Like TOS: Captain Soliloquy, Space Western, Captain/Science Officer team.
Unlike TOS: Very different mission, Romantic Relationships, the most serialized.
Voyager was perceived in its time as a step back towards TNG, which in some ways made it more like TOS and in others less.
We're on a ship again, and unlike TNG we're far from the comforts of known space, which lends a TOS air to things. Further, we have a Captain who gets very involved in decisions, and demonstrates knowledge of the technologies of space travel that Sisko & Picard probably considered outside their department, which makes her in some ways very like Kirk even as it makes her have the most casual way of talking we've seen yet.
Perhaps related to that casualness, we have a Captain who is doubted by her crew and has to win their allegiance more than any other. This is strictly relative, given that 90% of the time people assume Janeway knows best, but crew-integration bumps early on and the Janeway / Chakotay disagreements in hard cases like "Scorpion" are the closest the main character crew of any Trek series come to mutiny without some sort of mind control or impersonation being involved.
Voyager is even more like TOS in its all-crew cast. The many fatalities in the first episode and the remoteness from everywhere means that everyone is busy, In fact, since newcomers Kes & Neelix consider themselves crew-members, there's basically no non-crew until various child characters start to creep in in the later seasons. This, plus the split Federation/Maquis origins, makes the show a lot more about the stress of command and obedience than any show since TOS.
It could be argued that Voyager also takes a step back in the area of serialization, but I'd say it's mostly a step sideways. While they don't do long arcs, and development of the universe beyond the ship is limited by the one-way trip home, that trip puts a unitary focus on mission progress that even DS9 didn't have. You could practically have a "light years from home" at the beginning of every episode instead of stardates, and reducing that number is at the center of practically everything that happens. This is probably even more foreign to TOS than DS9's space operatic structure was (though of course having a lot in common with the past & future Battlestars Galactica and Battleships Yamato).
Romance is a mixed bag. (Tell me about it!) The one-way mission means we get more one-night stands for the Captain and crew, but the closed ship is a perfect incubator for Torres & Paris, who create the most believably developed "new" romantic relationship in all of Trek. But in terms of screen-time, the marriage problems are way down from DS9.
Voyager, in brief...
Like TOS: Hands-on Captain, Exploration, one-shot romances, crew tensions.
Unlike TOS: Plain-folks Captain, two people fall in love believably, all about getting home.
Enterprise is a strange case. It has the absolute least Kirk-like Captain (if you thought Janeway was plain-folks, or Picard was hands-off, Archer has them both beat), but is a big return to Trek in terms of its universe: Vulcans are a big deal again and we see the Andorians after a very long absence. (Klingons remain inescapable). On the other hand, they did lapse a couple times to problematical fan service for TNG groups like the Ferengi & Borg. We also lose Majel Barrett as the ship's computer.
The mission is very frontier-y like TOS, but has more of a sense where every little step is a victory over adversity, like Voyager. And you have long-building story arcs, like DS9 except maybe moreso.
One way that Enterprise is different from all the others is the absence of a character perpetually ready for love. Up to now there'd always been one, albeit with a steady regression through the ranks. In TOS it was the Captain, in TNG the first officer, in DS9 the Chief Medical Officer, and on Voyager the Helmsman. Maybe there is one on Enterprise, but he's too far down the ranks to appear.
On another crew-relations note, we're back to an ensemble dominated by the relationship of the Captain to his Vulcan first officer, with only the Doctor character really standing out on a regular basis among the others (although that's on his own rather than completing a TOS-like trio).
I'm going to leave off there, because although there are episodes of Enterprise I quite enjoyed, trying to make systematic sense of it makes my eyes drift away from the screen.
Enterprise, in brief...
Like TOS: Voyage of exploration, old school alien races, Captain & Vulcan dominate.
Unlike TOS: The least Kirkly Captain, big season-long plots, no Majel.
Putting it all together, I'd say TNG, which seemed superficially the most like TOS, is in fact the most like TOS. Much as with the origins of the Civil War or whose fault World War One was, the obvious answer is quite often the correct answer. But it's still worthwhile to look closer!