I'll be referring to & directly copying from Ariane DeVere's transcript, as I am a lazy bitch.
This will be edited a lot. Please let me know if I missed anything or messed up somewhere. The actual cut bits will be in bold.
1) After the scenes with the Geek Interpreter. The following two scenes involving 'The Speckled Blonde' (at the morgue, and naming the case at the flat) were cut.
2) During the scene where John is writing the "Sherlock Holmes Baffled" post. The scene ends after John's line "This is your living, Sherlock" - the rest of the line ("not two hundred and forty different types of tobacco ash"), as well as Sherlock's response ("Two hundred and forty-three") was cut.
3) After Sherlock and John laugh, after arriving at Buckingham Palace. After John's line "At Buckingham Palace, fine" - the line "Oh, I’m seriously fighting an impulse to steal an ashtray" is cut.
4) When they're discussing Irene at Buckingham Palace. After the line "There are many names for what she does. She prefers ‘dominatrix’" - the following exchange is cut:
MYCROFT: Don’t be alarmed. It’s to do with sex.
SHERLOCK: Sex doesn’t alarm me.
MYCROFT: How would you know?
5) As Sherlock and John leave Buckingham Palace. After the line "Text me the details. I’ll be in touch by the end of the day," the rest of the scene (involving deductions of the Esquerry and borrowing his matches) is cut, as well as the following scene in the cab (discussing the stolen ashtray.) It instead cuts immediately to Irene flipping through photographs of Sherlock on her phone.
6) As Sherlock is first standing in front of Irene's safe (immediately after the CIA agents encounter John). The following lines of dialogue are cut: Hmm. Should always use gloves with these things, you know. Heaviest oil deposit’s always on the first key used – that’s quite clearly the three – but after that the sequence is almost impossible to read.
7) As Mycroft is meeting with Sherlock and John at 221B. Immediately after Sherlock's line "She’d applaud your choice of words," the following exchange is cut:
SHERLOCK: You see how this works: that camera phone is her “Get out of jail free” card. You have to leave her alone. Treat her like royalty, Mycroft.
JOHN: Though not the way she treats royalty.
8) After Sherlock leaves the morgue. The entire scene following that (involving Mycroft calling John, them talking about the danger night, Jeannette dumping John, Sherlock returning and mentioning his sock index) is cut. It instead goes straight to Sherlock playing his violin.
9) After Sherlock resumes playing the violin (after unsuccessfully attempting the code '1895'). The rest of that scene is cut, including the exchange between John and Mrs. Hudson.
10) As John is talking to the woman outside. After the line "So, any plans for New Year tonight?", the line, the following exchange is cut:
JOHN: Er, nothing fixed. Nothing I couldn’t heartlessly abandon. You have any ideas?
11) After Irene shows up at 221B. After the exchange "So who’s after you?" / "People who want to kill me." / "Who’s that?" / "Killers," the line It would help if you were a tiny bit more specific is cut. Also, after John tells her they don't have the camera phone ("We're not stupid"), Irene's line "Then what have you done with it?" was cut.
12) When Sherlock is asking for the camera phone. After he says "I'll still have it," John's next line "I’ve gotta give this back to Mycroft. You can’t keep it," as well as the following shot of Sherlock's outstretched hand, was cut.
The following are cuts to the nonverbal scenes, the length of a scene, or the transition of a scene. I'm only putting in significant cuts. There were a number of scenes that had less than a second shaved off of extended shots - they did this a lot. I'm not listing those. ('cause this list is way too long already.)
1) After the “Hatman and Robin” incident. The PBS airing has Irene saying her line (”Hello. I think it’s time, don’t you?”) over the following shot that pans over Baker Street. They also cut the inside shot that pans over the mantel, Cluedo board, and Mrs. Hudson taking the gallon of milk. Instead, it skips ahead to Mrs. Hudson in the kitchen.
2) After Mrs. Hudson shouts “Boys! You’ve got another one!” Her little whimper of distress is cut.
3) They cut about a second of the young officer talking to the man inside the car (right before he says the line ”Sir, this gentlemen says he needs to speak to you.”).
4) After the officer says “No, sir – the helicopter,” we see a shot of the helicopter landing. The following shot, showing John and the officer watching it, was cut. They also cut a second or two of the following shot showing the man bringing Sherlock his clothes.
5) A second of the scene when John arrives at Buckingham Palace (which mainly showed the man guiding John in the palace walking away from the camera) was shaved.
6) One of the shots in the Palace of John walking to Sherlock on the couch (the view from behind the couch directly opposite Sherlock) was cut.
7) Now this is interesting. After Sherlock asks where Irene is, Mycroft says ”Uh, in London currently. She’s staying ...” and Sherlock stands up and replies ”Text me the details. I’ll be in touch by the end of the day.” In the original scene, as Sherlock is standing up, the shot during those two lines is set through the view of the door, perpendicular to the couches. However, in the PBS edited version, the view of Sherlock standing up is from behind the couches – a shot that does not exist in the original version. (Probably to make the following cut, which goes straight to Sherlock saying “Laters,” seem more natural.)
8) After John checks Kate, the scene of checking the bathroom to the bathroom is cut.
9) After Sherlock puts the phone down after Mrs. Hudson asks him to turn it down, there’s a shot of John, then Sherlock, that is cut (before John says ”I’m wondering who could have got hold of your phone, because it would have been in your coat, wouldn’t it?”).
10) As Sherlock goes to the window playing “God Save the Queen,” the shot transitions to winter and Sherlock playing “We Wish You a Merry Christmas.” In the original, the shot stays on the transitioned window for a bit before Sherlock appears; in the PBS version, it just cuts straight to when Sherlock appears.
11) After Mrs. Hudson says ”It’s the one day of the year where the boys have to be nice to me, so it’s almost worth it!” - there’s a shot of Molly staring at Sherlock which was cut.
12) After Sherlock says ”I see you’ve got a new boyfriend, Molly, and you’re serious about him.” Molly’s little gasp before she says “Sorry, what?” was cut.
13) After John asks Sherlock if he’s okay, and Sherlock responds ”Yes” - the PBS version cuts the scene immediately after Sherlock closes the door. The rest of the scene showing Sherlock’s reaction was cut, as well as the following scene showing Mycroft looking out the window. (The shot of Bart’s is also slightly shorter.)
14) After Mycroft leaves the morgue, and Molly looks at the body, it cuts a bit off that scene, as well as a bit of the shot looking at Sherlock through the door. (It instead cuts straight to Mycroft opening the door.)
15) After Sherlock tries “1895” and says the phone is just faulty - the next shot of John shows him looking away, then looking back at Sherlock. For some reason, John looking away is cut.
16) After John walks away, the extended shot of Sherlock playing the violin is cut (as well as the following scene with John), going straight to Mrs. Hudson in the doorway after John has already left.
17) Quite a bit of Sherlock’s reaction before he goes up the stairs to get Mrs. Hudson was cut.
18) After Sherlock headbutts the CIA agent and checks Mrs. Hudson, he turns back to look at the man. When he starts to stand up – as well as the next shot of the car pulling up to Baker Street – is cut. It goes straight to John approaching the front door.
19) A tiny bit of Mrs. Hudson looking up at Sherlock before saying ”You left it in the pocket of your second-best dressing gown, you clot” was cut.
20) After the scene with Mrs. Hudson, a tiny bit of John getting a glass from the kitchen is cut.
21) At Bart’s, after Sherlock tries 221B, he clicks the phone, exasperated. The rest of the scene (Molly looking at him, and him sitting down at the computer again) is cut.
22) After Mycroft receives the text from Moriarty, there are three shots pulling away from him as he panics. (Well, “panics.”) The middle shot of that was cut, and the last shot of Mycroft rubbing his head is cut considerably shorter.
23) The scene of the car approaching the jumbo jet is much shorter.
24) After Mycroft says the line ”I also don’t frequent cafes,” it cuts a bit of Mycroft closing his umbrella, and cuts the shot from above of him picking up the suitcase and going inside, with John following.
25) After Mycroft pushes the file to John Watson, a bit of Mycroft putting his resting his head on his hands is cut.
26) After Sherlock says he doesn’t want to look at the file, there’s a considerable amount of John staring at Sherlock that was cut out (before he says ”Listen, actually…” and Sherlock asks for the camera phone.)
Cuts made to "The Hounds of Baskerville"