The difference was that rather than turn it into soap opera porn, the characters actually tried to behave professionally (most of the time).
• DS9 only had a little soap-opera drama, serving as almost a kind of fanservice sprinkled judiciously into otherwise-thematic B-plots.
• VOY had more, with entire B-plots of episodes now being character-focused, including a few arcs that would have worked just as well if transplanted into some modern-age military drama with no SF elements.
• Drama-as-the-A-plot episodes were appearing by ENT.
(After that I stopped watching, but I've heard the trend has just continued from there.)
I really enjoyed TNG as well, and DS9 probably wouldn't have even happened if not for TNG's excellence. I never really felt like the writers gave Picard's character a lot of room to change, though. A lot of the time, he was almost like the sitcom dad who was mainly there to teach the kids a lesson at the end of the episode. That's not a knock on Stewart's acting ability; I'd say he was almost certainly the best actor on the show.
Also agree about Bashir.
And some of O'Brien‘s best performances were counterpointing Bashir too.
I think as a solo character, he’s a little shallow. But he works exceptionally well at being an anchor for other characters to play off.
But yes, those instances were a bit rare.