The bug

As mentioned in the title, non-op players can run op commands from signs (using "run_command" "clickEvent")  but not from written books (also using the "run_command" "clickEvent" mechanic). If that's not a bug, it's definitely inconsistent. IMO players should be able to run op commands from books as well. Makes the life of map makers much easier.

How to reproduce

1. Create a new world with cheats disabled
2. Open the world to LAN with cheats enabled
3. Give yourself a written book with a "run_command" clickEvent:

   /give @s written_book{title:MC-201807,author:Author,pages:['{"text":"Test","clickEvent":{"action":"run_command","value":"/give @s stone"}}']}

4. Close and reopen the world
5. Open the book and click on the word "Test"
   → The command fails (you are not given a block of stone)
6. Open the world to LAN again with cheats enabled
7. Open the book and click on the word "Test"
   → The command runs successfully (you are given a block of stone)

Note

In the above example, if "/give @s stone" is replaced with "not a command", by clicking "Test", "not a command" will be written in the chat, as if the player wrote it, which explains the above behaviour. It doesn't really execute a command, it writes in the chat instead. Signs don't do the same, so by clicking this sign :

/give @p minecraft:oak_sign{BlockEntityTag:{Text1:'{"text":"","clickEvent":{"action":"run_command","value":"not a command"}}',Text2:'{"text":"Test"}'}}

, nothing will happen.