GTD. Neutral ground to post this in.
It's been two years since I've seen SNW. I've seen every episode of S1 exactly once and not even all the S2 episodes. Nothing from Season 3. But it sounds like they've made Spock very emotional.
As far as Spock in "The Cage", the most you can is that at times he SPOKE IN ALL CAPITALS, probably to get someone's attention if something's important. KIND OF LIKE WOLF BLITZER ON CNN. He does the same in "Where No Man Has Gone Before" and "The Corbomite Maneuver". He smiled once in "The Cage", which was a huge mistake compared to the way the character behaved everywhere else in TOS. In "Where No Man Before" he says outright that he doesn't feel and all he knows is logic.
The overall arc of Spock in TOS is that he tries to be as unemotional as possible. Gradually, over the course of the three seasons, he loosens up a little. Except for some hiccups like "That Which Survives" where he's super-rigid again, but I blame that on the writer.
Overall, though, he's unemotional. To quote McCoy in "The Menagerie", "If [Spock] were caught acting like us or even thinking like us, it would completely embarrass him." As happens in "Amok Time". He has that huge outburst, then tries his best to pretend that didn't happen.
Also, Chapel was pining for Spock in TOS, whereas it seems that in SNW it's the other way around.
"Well, in SNW we're seeing Spock younger!" Well, we saw this younger Spock in "The Cage". See what I said above. I don't know how you get from Spock in "The Cage" to emotional lovey-dovey Spock in SNW.
"Who cares about continuity? It's the story that matters!" Yes and no. Things like, "The Enterprise can goes at Warp 9 in that episode when they couldn't go passed Warp 7 in that episode!" Yeah. I agree. Who cares? I'm not talking about the extreme stuff that people like James Dixon were obsessed with.
What I'm talking about is following character development. If you like following Spock's development in SNW, that's great. And you like following Spock's development in TOS, that's great too. If you put the two together, they don't go together. It's like going from emotional Spock who wears his romantic feelings on his sleeve to a Spock who represses feelings and who 99% of the time romantic charms aren't going to work.
At best, all they'll do is come up with some sort of forced explanation for how SNW Spock becomes like TOS Spock, assuming they even try.
Now, I haven't seen SNW in two years, but this is what I remember of him.
I blame this more on the writing than on Ethan Peck's portrayal. In DSC Season 2, where he has some existential crisis, it's explained by he's having a mental episode when we first see him in "Light and Shadows". He gets over better by the end of "If Memory Serves" more or less, even though the sight of Burnham triggers childhood inner turmoil. And then suddenly emotion he's not used to experiencing comes to the surface. It's unusual for him. It's not his norm.
As far as Zachary Quinto's Spock? I liked him in the 2009 Film. A lot better than Kirk, in fact. Less so Into Darkness when he screams "KHAAAAN!!!!!" and then goes full-on Incredible Hulk on Khan.
.....those are classics. How could you???? 
