When Burnham and Spock land on Talos IV the computer says the atmosphere consists of 4.6% CO2, which is lethal to Human life. So why wouldn't Burnham wear an EV suit to go out in that harsh climate? Talos IV is said to be an M-Class planet meaning it is sustainable to humanoid life which would mean that the levels of CO2 are below that amount of concentration. Clearly an error on the show runners they should have put it at .46% CO2 not 4.6%.
In the original series, the Talosiams were played by women, and their heads pulsed in and out when using their their powers. In DISCOVERY, the Talosians are played by both men and women, and there is no visible movement to their heads when using their powers. (Also, the modern makeup includes a nose piece, a "trick" that has been used in practically every series since STAR TREK: THE NEXT GENERATION to signify non-human species.)
Information on Talos IV is supposed to be classified to prevent anyone from attempting to exploit the abilities of its population, yet Burnham is able to pull up information on the planet, with explicit reference to the native inhabitants' mental abilities.
Pike is surprised to see Spock smile when he arrives in Discovery's shuttle bay. In the footage from The Cage (1966), played as a recap at the beginning of this episode, Spock smiled in Pike's presence.
The first officer of a starship is responsible for the conduct and discipline of the crew. Saru allowing the fight between Culber and Tyler to play out violates the rules and morals of his position.