On a 1-10 scale of science-fiction geekdom, I give myself a 5. I know not to call the genre "sci-fi," for instance. I know Margaret Atwood once said something about squids in space that didn't go over well. I once read half a Vernor Vinge novel and quite liked it, until I became unspeakably tired of it.
So I now offer these thoughts, knowing that the series in question are both so monstrously popular and intellectually picked-over that I may largely be revealing my own relative newbie-ness. Nevertheless. In the past month or so I have become completely obsessed with the new version of Dr. Who. (I never watched the old version; growing up, I knew it as just some odd thing that was apparently on from midnight till morning on the PBS station.) I've gone out of sequence, starting with the first season of Matt Smith, then going back and doing the three David Tennant seasons, and then backing up all the way to the single first season of Christopher Eccleston. (This first incarnation has been rightly judged as not great, especially in comparison to the others. But it wasn't really Eccleston's fault. They were starting an old series over, and learning as they went what worked and what didn't--for example, that the Doctor should not resemble, in the show's own words, a U-boat captain.)
So here's what I've observed. If Dr. Who is as huge, or huger, in the UK as Star Trek is here, the two series seem to reveal fundamental differences in our respective national self-perceptions. While I like most of the various iterations of Star Trek just fine, I've always found them to be fundamentally dull. That's because the characters are fundamentally dull. From Next Generation on, the human characters are not allowed to be flawed in any serious way. They may have some superficial "quirks" (like blindness or, I dunno, a nagging sense that one hasn't measured up to one's father)--but any real flaws (greed, violence) are offloaded onto other races, who are then mocked and/or fought. This whole structure seems to track with the doctrine of American exceptionalism, in which we (Americans/humans) are always the good guys, whose good intentions always, in the end, produce good results. Because we're good, nothing can really go wrong.
In contrast, might Dr. Who represent the emotional realism of a former empire? Because in this series, the "good guys," while making us love and root for them, do not always do the right thing, for various reasons. Often the right thing to do isn't clear, or it's a choice between two or more equally troubling options. Also, because Dr. Who is focused on time travel, we can discover that what appears right at one moment turns out to cause problems far down the line--as when your apparent rescue of a civilization turns its members into TV-addicted bozos hundreds of years in the future. Your attempt not to cause someone pain causes them more pain, because it turned out you weren't thinking from their perspective at all, even though you told yourself you were. Sometimes, you are just kind of a jerk, because you are tired and stressed and lonely and jealous, like everyone else is.
Perhaps it's un-American of me to prefer this viewpoint. Perhaps it's even defeatist to think that every solution you create might also create problems. But--this is crucial--this does not imply that one shouldn't bother. Knowing better than anyone how things can go wrong, the Doctor never gives up trying to save the universe, and never gives up on humanity, either. He loves humanity. In this way the show might even be more optimistic that Star Trek, because it allows for real human flaws to play a part in the story--even to be the story--while still advocating for the good fight. It's about doing your best, even while you know that time and entropy and your own failings will undo your deeds.