BBC
There's a lot of debate among the "Doctor Who" fandom over which showrunner is the best at Christmas specials: Russell T. Davies (who rans seasons 1-4, and returned as of season 14) or Steven Moffat (who ran seasons 5 through 10). The debate is made simpler by how showrunner Chris Chibnall (seasons 11-13) inexplicably decided to make New Year's Day specials instead, so this debate only comes down to Davies v Moffat. They both have different approaches to the format of a Christmas episode, so a lot of this comes down to simple taste.
Davies took a pretty dark approach. His first Christmas special, "The Christmas Invasion," revolved around aliens who tried to get one third of humanity to throw themselves off of buildings, and it ends on the surprisingly brutal note of the new Doctor killing Harriet Jones' political career after she blows the fleeing aliens up. The Doctor vs. Harriet dispute is a messy, bitter one with no clear answers; there are zero warm fuzzy holiday vibes here. Davies' specials may technically take place during the holidays, but they could all easily be reworked as a regular episode without much effort.
Meanwhile, 2010's "A Christmas Carol," Steven Moffat's first holiday special, is Christmas-y to its very core. The whole thing is inspired by Charles Dickens' novella of the same title, with the Doctor serving as the ghost of an old greedy grouch's past, present, and future. The whole thing is very sappy and sweet, playing out like a whimsical fairytale. Most importantly, it ends on a happy note, with lots of singing and hugging and snow. Is the episode as strong as the Muppets' take on the same Dickens' story? Not quite, but we give it points for trying.
It's no surprise that "A Christmas Carol" is rated 8.5 on IMDb, and is now the highest-rated Christmas episode in the show's history. (Well, there's one exception — scroll down a bit for that.)
A Christmas Carol is Steven Moffat at his creative height
BBC
So, what makes "A Christmas Carol" so great? Well, there's the fact that the kid actor playing young Kazran, Laurence Belcher, is actually good — a rarity for child actors on "Doctor Who." But mainly it's because of how clever and inventive it is. Moffat's timey-wimey spin on Dickens' famous tale is inspired, especially that final twist where old Kazran finds out that he is the Ghost of Christmas Yet to Come, being shown to his child self as an example of a future to avoid.
Admittedly, there's the plot issue that the use of time travel in this episode technically makes no sense. It brazenly breaks a lot of the rules the rest of the show has long established. But hey, it's Christmas, who cares?
This special came out at a time where showrunner Steven Moffat seemed on top of the world, where the cracks and repeated tropes in his writing hadn't yet started to show. "A Christmas Carol" promised great days to come, and when it comes to Christmas specials at least, that promise mostly came true. 2012's "The Snowmen" was one of the best companion introduction episodes the show's ever given us, and 2015's "Husbands of River Song" was perhaps the most romantic story of Moffat's already romance-heavy era. 2011's "The Doctor, the Widow and the Wardrobe" was also an episode.
Admittedly no Moffat Christmas special ever quite beat out "A Christmas Carol," but I'd argue that most of them were better (and more holiday-appropriate) than the specials from Davies' era. Then again, there was one RTD-era Christmas episode that technically beat out Moffat's best Christmas episode, at least as far as IMDb's concerned...
The End of Time: Part 2 is the show's highest rated Christmas episode, or is it?
BBC
"The End of Time," Parts 1 and 2 were both part of the 2009 "Doctor Who" Christmas special, although neither actually aired on the holiday itself. "Part 1" aired on December 26th and had an 8.2 rating on IMDb, whereas "Part 2" aired on New Year's Day and got an 8.9 rating. This means that, if you're inclined to include these episodes in the ranking, then "Part 2" is the true highest-rated Christmas episode of the series.
But date concerns aside, I don't think "The End of Time" deserves that honor. I believe the only reason it has that high of an IMDb rating is because IMDb in general tends to overrate finales, and "The End of Time" is basically the finale to the entire RTD era of the show. Like most of RTD's finales, there are a lot of half-assed plot resolutions and general over-the-top cheesiness, stuff that would drag down a regular episode's ratings if not for RTD tending to nail the emotional character beats. "The End of Time: Part 2" is a mostly effective, satisfying closing note for Tennant's Doctor to go out on, but that 8.9 rating is inflated.
For proof of the over-inflated finale phenomenon, just look at other widely-considered stinker finales in the show's history. Season 3's "Last of the Time Lords," for instance, was widely ridiculed at the time for its absurd anti-climax resolution, and yet it still has an 8.3 rating. Season 6's "The Wedding of River Song" was also very divisive, known as the moment where Moffat fully lost the trust of a lot of his fans, and yet it still has an 8.4. If a regular episode was this sloppy, you'd expect the rating to be somewhere in the low 7s, high 6s. You can see this trend on plenty of other TV shows too, like how the abysmal season 5 finale of "Shameless" still has an 8.2 rating, or how the famously-divisive "Lost" finale still has a 9.2. "The End of Time: Part 2" might technically be the highest rated Christmas episode on IMDb, but outside of IMDb "A Christmas Carol" still reigns supreme.