Async Business rule: Async business rule is same as after business rule and should be used when there is no urgency to update the records immediately and control should returns to the user.

Example: Close all child tickets when parent ticket is closed. As there is no urgency to close child tickets immediately Async business rule can be used so that control returns to the users.

Query BR: It always run on "before" database operation which restricts the user to access certain records.

Example: "incident query" BR on incident table is the perfect example which allow only ITIL users to access the incidents or if the incident is opened by or requested for the current logged in user.

Pls. mark this correct/helpful, if applicable.