Whenever Trek fans get on my case (and boy, they take this stuff seriously!), like clockwork they condemn Doctor Who for being the cheesier, cheaper show. Fair enough, guys. Doctor Who cost 1/5 as much as Star Trek and looked half as good.
Doctor Who takes the gold medal in every other event. It's better written, better acted, more moral, and more interesting.
Such statements are subjective, of course. I can but try to substantiate them. First, I'd like to cite a few examples of Who's superior scripting. Both shows tackled a lot of similar concepts, and Doctor Who inevitably handled them better.
Problem: you want to write a science fiction TV show with a God-like character, but you don't have a big enough budget to show the character doing anything really spectacular.
Star Trek writer solution: have the character teleport around the set with the aid of fast cutting. Give him a spangly robe that suggests power.
Doctor Who writer solution: convey the character's divine nature through carefully constructed dialogue. For example, when the Doctor introduces himself as a "Time Lord," have the god-like creature say, "A Time Lord...a lord of Time? Are there really Lords in such a small domain?"
A simple example, I know, but it's representative of Doctor Who's subtler, smarter approach to sci-fi. Here's another example. Doctor Who's most popular villains, the Daleks, were a race of identical, racist robot creatures that sought to purge the cosmos of all alien life forms. They all looked the same and talked the same, so they got a bit boring after a while. They needed a distinctive, charismatic humanoid leader to keep from becoming stale. The trouble is, since they're racist robots, the Daleks would never accept a humanoid leader.
Doctor Who's solution: They gave the Daleks a humanoid creator, Davros. Because Davros wasn't quite like them, the Daleks regarded him as inferior and tried to murder him. Davros escaped and created a group of less independent Daleks that were loyal to him. In one of the show's classic storylines, the rival factions of Daleks started a civil war over their split loyalties to Davros.
Star Trek encountered a similar problem when it "borrowed" the classic Doctor Who villains and created the copycat Borg (what kind of a battle-cry is "assimilate," anyway?). The Borg were identical-looking, racially superior warriors who sought to purge the cosmos of all other life forms (sound familiar?). Like the Daleks, they got boring and limited after a while. They needed a dynamic, humanoid leader. The trouble was, as a gestalt, communal entity, the Borg would never accept a single leader.
Star Trek's solution: give the Borg a Queen. Don't explain why the Borg accept this Queen when they're supposed to function as a group. Why bother to explain little things like character motivations and alien mindsets? There's thick technical manuals to write about the precise tachyon emission level of the hydrogen bussard collectors on the Enterprise's port warp nacelle; that's clearly more important than consistent characterization.
The Borg are silly villains for the Enterprise crew to fight anyway. All of the characters in Star Trek: The Next Generation think and act alike, just like the Borg do. They're all glad to live in a sterile future where ultra-liberal values rule, smoking is prohibited everywhere, and alcohol is synthetic. The Federation is comprised almost entirely of morally superior snots; they purport to be tolerant, but they turn their noses up at the Ferengi just for having different values than they do. The Doctor, a true citizen of the universe, accepts every nonviolent person he meets, no matter what they believe in. And as a true individual, the Doctor is the perfect choice to fight zombie-like clone villains like the Daleks and the Borg. The Federation are the Borg already. As one SFX magazine critic said, Star Trek's worldview is "humanistic fascism." Overstated, but essentially true.
Star Trek's fans and creators are fond of praising the show for being very moral, progressive, and ahead of its time. I don't really see it. Kirk nails every woman in sight in the original series. The crew is divided into 6 men (Kirk, Spock, McCoy, Scotty, Sulu, and Chekov) plus one woman, Uhura (the least important character). In one episode of the original series, we find out that women can't be starship captains because they're "too emotional." Right...
Doctor Who almost always featured a balanced cast of men and women, and the women, though slightly less important, always had something to do. Uhura's job was mostly to take the Captain's messages; Troi was kept around so she could show off her cleavage; Seven of Nine, the apex of political correctness, is considered a great character because she has breasts the size of weather balloons. No Aces, Barbaras, or Sarah Janes here.
Some of Star Trek's best writers have publicly admitted that Doctor Who is a better show. Denny Martin Flynn, who co-wrote Star Trek VI, said that he preferred writing a Who script because Doctor Who was a much more open, creative concept. Harlan Ellison, who wrote City on the Edge of Forever (the most acclaimed episode of Star Trek), told a convention hall full of people that Trek would turn their brains into "puree of bat guano," and recommended Doctor Who instead. Funnily enough, Robert Holmes, Doctor Who's best writer, did not prefer Star Trek. He thought Trek was inferior too.
I could go on and on (I already have!), but I'll just summarize. Star Trek is less intelligent than Doctor Who, but preachier. It's more expensive, but less visually interesting. It's more self-congratulatory, but with less reason. And it hasn't got Tom Baker.
Star Trek and Doctor Who are the most enduring, universally popular science fiction shows ever made. Trek dominates the American market (we Yanks have no respect for any foreign entertainment, believe me), but on the worldwide stage Doctor Who has won over 100 million fans in 80 countries. Obviously, both programs have their respective merits; junk doesn't continue to sell for 25 years. But one of them's got to be better than the other...