The "Approve tag wiki edits" privilege is currently awarded at 5,000 reputation.
However, the ability to edit tagwikis without needing approval is part of the "Trusted user" privilege, which isn't awarded until one reaches 20,000 reputation.
This creates the odd situation where a user with 5,000-19,999 reputation can approve suggested tagwiki edits, yet their own tagwiki edits themselves require approval by others.
In contrast, for questions and answers, the ability to edit without needing approval and the ability to approve suggested edits are both awarded at the same reputation level (as part of the "Edit questions and answers" privilege, awarded at 2,000 reputation).
Why are the reputation levels required to (on the one hand) approve tagwiki edits suggested by others and (on the other hand) edit tagwikis without needing approval so vastly different?