One of the key differences I have noticed between the Davies Era and the Moffat Era is based on a simple question: Who is the central character of the show?
Davies Era: For this era of the show, it is pretty clear that the companions are the central characters. The first scene of Rose features the introduction of Rose Tyler, rather than the Doctor. When Martha Jones is introduced in Smith And Jones, again the first scene features Martha, and not the Doctor. Even though Donna and the Doctor previously met in The Runaway Bride, the first scene of Partners In Crime features Donna Noble, and not the Doctor.
It’s accurate that not all the episodes featured all the new companions in the opening scene, as this occurred in Voyage of the Damned, The Waters of Mars, and The Next Doctor. The Doctor is present in all the first scenes of these episodes, but also none of the companions are seen past the episode they are introduced in (Astrid Peth, Adelaide Brooke, Jackson Lake).
This is also indicative of the series finales, as well. In The Parting of the Ways, the Doctor had sent Rose (with the TARDIS) back to Earth in her own time. In order for Rose to get back, she opened the heart of the TARDIS and took in the Time Vortex. Since the Doctor was incapable of defeating the Dalek Fleet in 200,100, Rose-as-the-Bad Wolf Entity removed the entire fleet from existence. After, the Doctor kissed Rose in order to remove the Time Vortex from her (because, you know, it would only be so simple).
Equally, in Doomsday, in order for the Doctor to defeat both the Daleks and the Cybermen, he needed someone (Rose) to keep the Void open, with him, on the other side of the room. After Rose gets trapped on the other side, the Doctor uses a holagram in order to contact her.
In Last of the Time Lords, as the Doctor was being held captive by The Master, it was Martha Jones who had saved the world by going around the globe telling everyone about the Doctor so that during the countdown, the Doctor would tap into the Archangel Network and reverse the Year That Never Was. The Doctor fixes the damage done by the Master and returns everything to normal, save the American President.
In Journey’s End, the Dalek Empire would not have defeated without the help of the Donna Noble as the DoctorDonna. When the Earth was brought back to it’s place in the stars, the Doctor, along with many other companions helped pilot the TARDIS back to it’s rightful spot.
Moffat Era: Although I did not catch this earlier on, Moffat’s era of Doctor Who, the central character in this era is clearly the Doctor.
Granted, The Eleventh Hour features the Eleventh Doctor in it’s first scene, but more importantly: the title is a direct reference to it being the Eleventh (incarnation of the) Doctor. This wouldn’t be the last episode to directly name him either, as there is also Vincent and the Doctor, The Doctor’s Wife, The Doctor The Widow and The Wardrobe, and The Name of the Doctor. Even minisodes directly name him with the five-minisode series called Night and the Doctor and most recently The Night of the Doctor, the prequel to The Day of the Doctor.
To say this is strictly been about the Doctor throughout the run would be not completely true. The Series 5 episode Amy’s Choice, has Amy Pond’s name in title, yet the foe revealed by the episode’s end: the Doctor. The Series 6 episode, The Girl Who Waited, refers also to Amy Pond, but the main foe in this episode is time. Another Series 6 episode, The Wedding of River Song, is directly referring to the companion of River Song, but who is she getting married to? The Doctor. Even more interesting, she didn’t even appear until at least mid-way through the episode.
It’s also true that not all of the minisodes feature the Doctor either. The five-websiode series Pond Life explicit mentions Amy Pond by name. At the end, Amy makes a desperate plea that she “needs” the Doctor though. There is also the second prequel to The Snowmen, Vastra Investigates: A Christmas Prequel, featuring Madame Vastra, Jenny Flint, and Commander Strax, though it is jovial in nature. There is also the minisode, Clara and the TARDIS, which features Clara in the TARDIS.
Regarding season finales during the Moffat Era, in The Big Bang, the Doctor creates Big Bang 2, resulting in the erasure of him from our universe. In order for the Doctor to return, Amy remembers him and the TARDIS back into existence.
In The Wedding of River Song, River Song refuses to kill the Doctor at Lake Silencio, resulting in all of time occurring at the same time. When the Doctor marries River, they kiss, and time begins to move forward again, taking them back to Lake Silencio, where the Doctor is actually inside the Teselecta.
In The Name of the Doctor, Clara send herself into the Doctor’s timestream in order to reverse the damage done by the Great Intelligence, saving the Doctor over and over again. After Clara completes that task, the Doctor goes into his own timestream in order to save her.
Even though during the Davies Era, the companions were still ultimately the Damsel in Distress that has placated the theme of Doctor Who for many years. However, it has had certainly somewhat more pronounced revival with the Moffat Era given that the Doctor’s name is in more titles than ever before beginning with The Eleventh Hour.
Coincidentally, the companions had a larger part of the story during the Davies Era, some of which remains, but not to the same degree during the Moffat Era. Because of this, I am suggesting that the Doctor is the central character of the show, and that during prior seasons he also was, but was at a different degree.