In the beginning you see Sarek preparing to leave Vulcan. In the background you can see two large moons in the sky despite the fact that Spock has said that Vulcan has no moons in the original series. This was answered decades ago (after the first Star Trek movie showed a similar sight): Vulcan has a sister planet (T'Khut) with its own moon (T'Rukh), which makes Spock's statement technically true.
When Sarek (James Frain) is healing In the sickbay and talking to Michael (Sonequa Martin-Green), the lace front of his wig is clearly visible.
Lorca and Tyler engage in a holographic battle simulation, despite the fact that this technology was introduced a century later (Star Trek: The Next Generation).