While good storytelling keeps a show alive, continuity keeps the universe the show is in seem at least a bit more real if it can maintain some semblance of consistency. Particularly if the show is long running and has multiple eras. That is not to say the show has to be a slave to its continuity. No. But either the writers or at least the art department should at least make an effort to not take the audience out of the subject matter by making things radically different without a logical explanation (be it in or out of universe).
The Klingons in Star Trek keep changing appearances every other decade it seems. Now the easy answer and the original answer was, that the studio could afford better makeup for the Klingons and went for it. So the Klingons would have always looked like the new appearance even when they did not. Until suddenly they didn't look like that because of the need to use decades old stock footage and new actors made up like those decades old actors to interact with the newer show content (Trials and Tribblations). Than, suddenly, there needed to be an explanation for this difference. "We don't talk about it with outsiders", is of course a funny copout, but it did mean there was an explanation. ENT gave us one about a decade later. Some people don't like it, but it was there.
Star Trek: Into Darkness gave use even more different Klingons, but no one really cared all that much. But now with Discovery, the Klingons are massively different even from that design. The makeup seems to have become too much that the actors, some what we can see so far, have trouble acting in the makeup. Maybe that was just for some of the actors and we'll see better later on. But this again is a massive change with little explanation. Now it can again be the older TMP explanation that, "that how the always looked" and we'll have to work with it, but the better explanation would be mutations, or one of a multitude of Klingon races within the Empire, or even a species that calls themselves Klingons, but either are not Klingons, or the species we've know as Klingons were never the true Klingons, just borrowed the species name for some reason and the Federation never figured that out (like if the Hur'q, as the 24th century Klingons called them, were actually the Klingons and what we know as Klingons took their name from their oppressors and made it there own, but wiped those records from their history. So they don't even remember what they called themselves all those millennia ago, and gave their former oppressors a new derogatory name.)
Now for Starfleet, we actually don't have a lot on it from the TOS time period. So the new ships and equipment can fit in or not as much as they like. The only thing that should likely remain the same, or at least closely to the same, is the USS Enterprise and her class of starships. If for no other reason than that is what people would expect to see. And even if it contrasts with the rest of the fleet, it standing out would still be there to say "hello, remember me?"
Doctor Who, for all its BBC cheapness back in the 1960s, still reuses 1960s props or likenesses to 1960s props for episodes that specifically relate to events from episodes that took place during that era of the show. In the 2010s, you can still see 1960s era Daleks and Cybermen used on TV along side modern versions of those creatures. The Ice Warriors outer armor remains more or less the same even though they updated the interior alien in appearance. The TARDIS is still a police box, though it make be better modeled on the real thing these days than it was in the 1960s. There are species they have updated and left updated. They have not returned to the old Silurians, and the Zygons, while still big red rubbery monsters, are better detailed now. The show will also reuse its 1960s era TARDIS interior....because they made one for the exhibit and can use it from time to time.
Star Trek can easily remake a 1960s era Constitution-class bridge layout with the colors and styling from the era, while updating the display areas and panels to reflect the rest of the fleet. It is not that difficult, nor is it out of the writers or art directors ability to wave off an explanation as to why this class of starships look like this. Different shipyard contractor is by far the easiest answer. The rest of the fleet was built in Andorian yards, the Constitutions were built in Human years. That is how simple they could make it.
