I see a lot of posts all the time that say RTD was by far the better writer, and there's some criticism still going on about the quality of Moffat's plots, and his emotional capacity.
Personally, I don't get the argument that RTD was a better writer. I like them both, and they both have strengths and weaknesses, but for me Moffat is the obvious choice between the two.
Early on especially I think a lot of Russel's plots were more goofy, almost cartoonish stories that were sometimes fun but also had a lot of cringe and unnecessary sappiness. He improved over time absolutely, and Waters of Mars and End of Time were both very emotional in my opinion. But outside of a couple episodes like that I don't think Russel has the emotional writing people make him out to have. And his plots normally range from decent to good, but then almost every ending he writes is poor, deus ex machina endings that come out of nowhere and cheaply resolve the plot (one of the worst examples being the Sound of Drums where speaking the Doctor's name somehow brings him back from being a shriveled Gollum thing).
On the other hand, I think Moffat has weaknesses, but overall he is a superior writer in almost every way. He brought a level of mystery and seriousness to the show and his plots are much more involved and better written in my opinion. I've seen criticism on Moffat's characters, and especially his supposed lack of emotions, but this is something I just don't understand. Compared to Russel his stories and characters are much more emotional. There have been many moments I've gotten slightly choked up over a Moffat plot, but that's never happened in any Russel story outside Ten's regeneration. Moffat writes overall better characters as well. If the Doctor is going to like someone it had better be someone like River, because she seems like his type and she isn't one of those companions that's constantly clueless and idiotic. Sure, she's not perfect but characters like her are what makes the Moffat era shine, honestly. The characters are a lot deeper and they are more involved with the plot than were companions in Russel's era.
Finally I think Moffat understands Doctor Who more than Russel did. I came to the modern show from growing up with the classics and in my eyes Moffat does a much better job honoring that legacy than RTD. Moffat did put spins on the series that wouldn't have been done in the old days but he writes the Doctor as he is supposed to be written and he stays true to Doctor Who being a true sci-fi as opposed to a soap opera with aliens.
One final thing I absolutely need to make clear: I like Russel's episodes. I don't think he's a bad writer at all. He's not my favorite writer ever, but that doesn't mean I don't think he has some very, very good episodes. Because he does. But if I had to pick between him and Moffat it's Moffat hands-down.