Nurse Chapel is pretty much an entirely different person, Spock has a wife who is part of his life.Smallville's a prequel to the idea of Superman, not the (then) Christopher Reeves, Superboy, nor Lois & Clark. The later stuff didn't exist yet. The same goes for Gotham, it was an origin story, but not a prequel to the anthology films nor the Dark Knight trilogy. BVS wasn't out yet. My understanding is that SNW was conceived of as and intended to be a prequel specifically to TOS, not just the idea of TOS. That said, I understand they'd take creative liberties for the 2020's audience. Surely it doesn't contradict the original show that badly, does it?
If you're into the technical stuff, the ship is much bigger and has far more advanced technology than the ship in TOS had, like a forcefield in the shuttlebay when the classic show had to wait for air to fill the shuttlebay after a landing.
As said, it's IMO a prequel in the same vague ways as the other shows I mentioned. In the eyes of Paramount, they act like it's a direct prequel, even if one episode explicitly acknowledges the timeline of the Eugenics Wars has been changed
The Eugenics Wars were moved up from the 1990's by Romulans, in SNW Khan Noonien Singh is a Canadian child in 2024.