Within the Special Demo are certain events that occur on specific days after you begin the demo. Most of these events are insignificant, but there are a few that might be worth your Time.
1 Day After Starting Demo: Speak to the man next to the Pokemon Center, and he will give you 10 Pretty Wings. These can be sent to your Pokemon Sun or Pokemon Moon game via the Pokemon Link.
5 Days After Starting Demo: Speak to the girl in the southwest corner of Hau'oli City. She will introduce you to another non-playable character that helped her when she had injured her ankle.
12 Days After Starting Demo: Talk to the man in the City Hall, and you will find that it's his Pikachu's Birthday.
18 Days After Starting Demo: Talk to the Police Officer in the Marina, and he will ask for your assistance with capturing some criminals.
24 Days After Starting Demo: There is a girl in the Marina leaving The Alola Region that needs some support.