What specifically does STC get right that NV/P2 doesn't seem to?
One thing, for what it's worth: nuance and polish.
Certainly STC isn't perfect--nothing is--but they do manage to recapture a lot of what made us like TOS. All fan productions diverge, in varying degrees and ways, from the original source materiel in bringing contemporary sensibilities to their productions. STC, like
Starship: Exeter, seems to do it less than most.
Fan productions are really elaborate forms of fan fiction--not a bad thing--but again it comes down to approach and execution. Fanfic (whether in print or filmed) is a case of wanting to play in someone else's sandbox. Some fanfic (okay, a lot of fanfic) seems to be a case of "wouldn't it be kewl if..." and it doesn't really have much resonance with what we would likely have actually seen on the series. I would even put a lot of professionally published stories in that camp given how so many writers like to bring back characters and revisit places and things that the series already did and likely would never have revisited.
The one thing that's inescapable (to me) about NV/P2 is that none of what I've seen in the series most likely would ever have been done in TOS even if the show had gone on for another two seasons. It's an approach the production has chosen to take to explore things that couldn't or wouldn't have been done in TOS either because of subject matter or budgetary constraints.
That in itself isn't what really what takes me out of the episodes (well, most of the time). In all candor what usually takes me out of the episodes are the writing, the acting and sometimes the look of the show. I've found the acting in NV/P2 to be all over the map, but mostly unconvincing overall. The cast just doesn't do it for me. Maybe if everything else were stronger I would cut them more slack, but as is when you're not getting enough enjoyment out of something then every little thing looms large.
Often a lot of the writing strikes me as almost rubber stamped. It doesn't seem to flow in a convincingly natural and spontaneous way. Now part of this might also be affected by the acting when a character comes off more like an impersonation rather than a fully realized character.
The look of NV/P2 also doesn't look like TOS for the most part. Here a lot of it has to do with the lighting and camera work. Then again because the production is doing something different so maybe it shouldn't matter. But it strikes me that if it looked more like TOS then so much else might seem more convincing.
STC has chosen a different and in some ways more restrictive approach. For the most part they want to seem more like it's 1969 again, for the most part anyway. They still do some things that most likely wouldn't have been seen in 1969 TOS, but overall they've usually gotten away with it mostly because they've exercised some restraint. Some of the visual f/x they've done actually could have been done in 1969
if TOS had had the extra time and budget. In particular I'm thinking of the hangar deck shots.
In terms of ideas they've also included some things that are debatable. It's very cool that they've shown us a woman Commodore and a woman as a security guard--both of which it would have been awesome to see on TOS. But strictly speaking we likely wouldn't have seen a female security guard in TOS and it's unknown whether we would have finally seen a female command officer in a fourth or fifth season. The inclusion of the holodeck (or holographic rec room) is an idea that existed back in the day and it's something that TOS could have pulled off (given all the things we'd already seen on the show), but they never got around to it.
The inclusion of a ship's counselor is the biggest debatable idea in STC. It's so much an element of the TNG era and one that we most likely would never have seen on TOS. Yes, they had had ship's psychiatrists, but they were one off guest characters and never part of the regular cast. I understand why they created the character, but it still can seem wierd. I would say, though, that again they get away with it because they haven't (so far) pushed her too much.
In terms of stories STC has walked a fine line (mostly) and crossed the line once in terms of story ideas likely to be seen in 1969/70. "Pilgrim Of Eternity" feels very much like it could have been a TOS episode (if we also accept them aging Michael Forest), but it's also unlikely a story that would have been produced given that revisiting a character was done only once before. "Fairest Of Them All" was pure fan indulgence because TOS was most likely never going to revisit the Mirror Universe. "Lolani" is a story that hits all the targets in terms of what we could have seen on TOS.
But even given where STC diverges from TOS in terms of ideas and story it is their overall execution that brings it all together. The show really looks like TOS in terms of lighting and camera work. The ship model is a an idealized version of what we saw onscreen even when the ship is seen in some angles not seen on TOS (but could have been). The stories are written and executed in a very close approximation of what we saw on TOS. Even though "Fairest Of them All" would never have been done on TOS it still feels very much like how it could have been done.
When it comes to acting while it can still be hit-and-miss on STC overall I think it's much better than NV/P2. No one is going to replace William Shatner, but Vic Mignogna's interpretation and performance of Kirk is generally awesome. Todd Haberkorn has and is growing as Spock. How can you beat Chris Doohan as Scotty?--it's spooky. The rest of the cast are competent so far given the limited scope of their characters. Michelle Specht has also been good and has managed to fit right in although she has an advantage of creating an original character without any serious fan expectations to overcome. I also have to say the guest actors have been very good with their characters.
So I think a key distinction is this (for me). Both productions do things that were not seen on TOS, but a key difference is that (for the most part) STC has given us things that
could have been done on TOS while NV/P2 has given us things that wouldn't or couldn't have been done. And additionally I feel STC is generally better written and better acted than NV/P2.
Of course your mileage may vary depending on what one wants to see in their fanfic.