Despite only being in the network for a matter of weeks in-story, Hugh's hair shows about a year's worth of growth, though his beard is not as long.
Hugh's hair was shown to already be graying in season 1, yet his overgrown hair (wig) and beard here are completely dark.
During Tillys "Thanks for that" line it is clear the shot has been flipped the star fleet insignia/division badge is located on the right side of the uniform when it is more commonly located on the left side of the chest.
The spores that attack Tilly's hand early on leave a visible burn mark, but the ones that attack her face/neck later do not.
If "May" could convince her fellow spore beings not to eat Tilly, she should have been able to do the same for Discovery.
Tilly announces that she is carrying a "type 3 phaser rifle." The type 3 phaser is the rifle configuration, with type 2 being the pistol and type 1 being the handheld version. Saying "type 3 phaser rifle" is redundant.