There's a truism that the big difference between US and UK sitcoms is that the former are gag driven, while the latter are character driven: viz. the difference between, say, Friends and Steptoe. This may or may not be (but probably is) connected to the difference in the way they are written: roomfuls of writers in the States, and single or pairs of writers in the UK.
I think the first half of that equation is true enough, at least as a generalization. That isn't to say that Friends et al don't have characters but that the shows are designed primarily as a series of pegs to hang gags on. It is only a generalization, though, and at times different cards can be shuffled to the front of the deck. I think for example of the two-hander episode of Family Guy (generally a very gag-driven show) in which Stewie and Brian spend the weekend talking in a locked bank vault. But there the writers seemed to be quite self-consciously "doing" that kind of episode, almost to show they could: it isn't their stock in trade. Whereas two-handers for, say, Steptoe and Son, were what the show was all about, and when they stepped outside that they had nowhere to go but the land of slapstick, where they were generally less successful, at least in the gentle art of Entertaining Steepholm, which is the highest end of all comedy. One could say similar things about Hancock or even Last of the Summer Wine, which in its early years consisted very largely of three old codgers wandering aimlessly about the Dales, putting the world to rights. The runaway-trolley silliness with which the show became associated took over only gradually.
I'm not sure that modern British sitcoms are character driven or gag driven, though. Nor are they even interested in a wide variety of situations. Especially since The Office, they seem to thrive specifically on creating theatres of embarrassment. Gags aren't at the heart of it: The Office was faux-documentary, which meant that gags - at least good ones - had no place at all, while one of the most successful sitcoms of recent years, Outnumbered, depended largely on letting its enfant terrible actors improvise cringe-making dialogue. Yes, there are characters in these shows, but if they are sympathetic and recognizable, that's largely because we need to empathize with them in order to feel their embarrassment effectively, rather than because the show's about them in themselves. They are enough like us, or like people we know or have been, for us to project ourselves into their situation. (I suspect this is what Aristotle wrote in his lost book on comedy - in which case, I agree with him.) Many more recent shows work on this formula, or perhaps I should say with these priorities, from The Inbetweeners to more recent outings such as Gates, but even something like Fawlty Towers wasn't character driven in the way that Hancock was - it was all about the embarrassment.
I wonder if this helps explain the relative dearth of historical sitcoms? I really can't think of that many. There's Blackadder, of course, and going back a bit there's Up Pompeii!, that curious mixture of New Comedy and seaside postcard, but that's about it. They tend towards the slapstick, anyway - as of course did the various historical Carry Ons at film length. Perhaps this happens as things pass from personal memory into history proper, and the chance for personal recognition disappears. Think of the difference between Dad's Army (began 1968) and 'Allo 'Allo (began 1982). The first is character driven, albeit with the same caveats as for Steptoe and Hancock: the parts that make the show worth watching are the ones where not much happens, but as soon as a large prop makes its appearance we know that a pratfall into slapstick will swiftly follow. 'Allo 'Allo by contrast is deliberately stylized and pinned together with catch-phrases (not that Dad's Army lacked those). The first was written for an audience that still, by and large, remembered the war; fourteen years later a demographic tipping point had been reached, the war was history, and the style of its comedic representation changed accordingly.