the Doctor is far more willing to "interfere" to correct an unjust situation, and I say good for him. The post-TOS PD is really on of the more problematic parts of Trek. Both shows have extremely open formats that allow for almost any kind of stories to be told, which is perhaps why they're two of the best shows ever. DW is "let's hop into my TARDIS and go wherever!" and Trek is "hop onboard the USS--- and let's go wherever." (DS9 the exception of course, but they still had runabouts and the defiant.)
Federation (and no doubt other political entities') facilities can appear bigger on the inside than on the outside by use of holotechnology; the TARDIS is the real deal. Plenty of sex/romance on TREK; next to none in DW, and close to zero for its main character--very unlike TOS in this last, anyway.
Well, it's not like Doctor Who is prudish or anything, at least not these days. And at least Doctor Who acknowledges homosexuality exists, hell it even featured a bisexual character who went on to star in his own spin-off. Star Trek uses lesbians to show the Mirror Universe is a different place.
The new series has had a lot of romance for the Doctor and often, his companions (For instance the Moffat/Smith run featured a married couple: Amy and Rory as his companions), although it sort of dances around him being in a fixed relationship. This started oddly enough with the Paul Mcgann TV movie. The old series though didn't really have much romance for the Doctor (apart from "The Aztecs") and the companions did fall in love, it was usually with the guest character in their last episodes (Such as Jo and Leela).
I remember I had the notion in the Tom Baker days (when, like many Americans, I was introduced to the Doctor) that maybe, at heart, The Doctor regarded humans as not too different from monkeys and was a little repelled at the idea of a liaison. We could argue that he changed after several regens (Rose, River). Or maybe just got horny enough. Side note: would you believe I was just watching "The Golden Voyage of Sinbad" this weekend? Every now and then I like to revisit 7th Voyage, Golden, and Eye of the Tiger, mainly for Ray Harryhausen's stop motion. Always kind of cracks me up to see Tom Baker hamming it 1 year before he starts as the Doctor.
I can't believe nobody's mentioned the Potato People yet - Trek's Hierachy and Who's Sontarans. Oh, and any fans of TOS and Tom Baker's Doctor might want to give The Doctor and the Enterprise a read. A fun crossover story which posits that Trek and Who diverged when in the distant past of one universe Vulcan died, and Gallifrey in the other.
I love both, but I really can't compare them. For me, they are completely different and give me different types of entertainment.
One of his first companions was his "granddaughter," and as far as I know, they've never gotten around to explaining that!