Who has access to what mod tools?
Trust Level 1
Ability to flag posts
Trust level 3
All of the above, plus Rename topics, recategorize topics, hide a post by a TL0 user by flagging as spam, Auto-silence TL0 users by casting enough flags.
Trust Level 4
All of the above, and grants many more: Edit all posts, take common topic moderator actions such as pin, close, unlist, archive, split, and merge.
Moderators can do everything in trust level 4, plus:
- Process flags
- Process posts held in the moderation queue
- View private user information, including email addresses
- Suspend and silence users
- Adjust a user’s trust level
- Delete topics
- Delete and anonymize users
Can do everything above, plus:
- Read and moderate any user’s personal messages
- Ignore category permissions to view private categories
- Create and delete categories
- Change category permissions
These are in order. Because moderators have every permission that trust level 4 has, there is no need to grant moderator without trust level 4, other than denying the user the badge. Admins have every single permission, and that's also the highest trust level.
(mentions: @Thanatermesis (who said he doesn't know a lot about discourse))