This is an important distinction between Star Trek and Doctor Who or, really, other British SF series--in my entire lifetime, the longest gap in live-action Star Trek has been the one we just had, 2005-2009.
You young whippersnapper. I was around for the ten-year gap in live action Star Trek, though I didn't get into the show until a couple of years into that decade.
The differences between the two fandoms are interesting, and I've long wondered why the two fandoms are so vastly different. The conclusions I've come to:
1) Doctor Who has no Roddenberry-like figure.
Everything else in the post is gold, but basically, this is it in a nutshell: there's no mythical Creator of Doctor Who. There's no Authority. And that parallels what the shows are about. One's about a more or less military organization, part of a hierarchical structure with clear lines of authority; the other is about a guy who dislikes all that stuff and just wanders around.
Star Trek fandom seems predisposed to need authority (all those "Gene Roddenberry would never have allowed this to happen" posts from people who clearly don't know how little core Trek stuff was created by Roddenberry) and canon. Doctor Who fandom seems to manage nicely without it. An oversimplification, perhaps, but I think there's some truth to it.