What are the characteristics that define a Whovian and a Trekker? Are you neither, either or both? I am both. But I acknowledge there are significant differences and fascinating similarities.
I tend to see Whovian’s as indulging in more convoluted thought – as an example when thinking of (or perforce diagramming) the complexities of time and the inevitable paradoxes that arise. Also I think a Whovian is more likely to think of problem resolution in terms of social engineering while a Trekker may look to a technological solution.
Dr Who is at heart a rationalist and fundamentally a scientist and I think a pacifist (as needed). Although his “diplomatic” style can be very confrontational. Conversely Star Ship captains, at their best, are like philosopher-kings or philosopher-warrior-kings. Not forgetting occasional bouts of “Cowboy Diplomacy”. Still I think they too are at heart rationalists and at least technocrats if not always true scientists. So the similarities regarding rationalism when engaging the universe should appeal to both Fandoms. Where they diverge is in the heavy reliance on technology (and I think a greater tendency to violence) in the Trek universe versus that of the good Doctor.
But if the TARDIS isn’t high technology, what is? Surely it is, just not the same as a Star Ship festooned with weapons. This dichotomy is also underscored in the difference between reaching for your sonic screwdriver instead of your Phaser. Really I think this difference is between being British or American. How often do we hear the Doctor tell his companions “Run!”? Probably just as often as we hear someone on Star Trek yell “Fire!”