Strange New Worlds actually are quite faithful to old TOS canon, mostly trying to circumvent tight spots, not outright contradict them.
The Gorn would like a word.
In TOS, ten years after SNW, the Gorn were unknown to the Federation. In SNW, ten years before TOS, the Gorn and the Federation are basically at war. There have been many well documented contacts between the UFP and the Gorn.
Canonically, the Klingons in this era should look like swarthy humans. The forehead ridges do not exist just then. "Enterprise" rather magnificently explained the issue, and DS9 confirmed it.
The SNW Enterprise is not only a different configuration than the TOS Enterprise, it's also something like 33% bigger (385 m long vs 289).
In TOS, Spock is an iconic character. In SNW, Spock is... not Spock.
In TOS, the Eugenics Wars happened in the 1990s. In SNW, thanks to Romulan time travel chicanery, they don't happen until the 2030's, earliest.
In TOS, Starfleet looks like a reasonably progressive military outfit. In SNW, somehow the gender roles common to humans for all of recorded history have been reversed and there seem to be more women in the armed forces than men.
And then there are the vast visual differences. As both DS9 and ENT showed, the original bridge design works just fine as a modern set, with just some minimal updates to displays and such.
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