scribu

Scenario
You’re the only user who can create posts and moderate comments on your WordPress installation.
You have both ‘Email me when anyone posts a comment’ and ‘Email me when a comment is held for moderation’ options checked.
You get an email about a comment awaiting moderation and you approve it.
The problem is that right after that you get a second email letting you know that a new comment has been posted, which of course you already knew about.
Solution
Instead of disabling email notifications for new comments alltogether, you can use this plugin and not receive the redundant emails mentioned above.
Links: <a href="http://scribu.net/wordpress/filter-email-notifications" rel="nofollow ugc">Plugin News</a> | <a href="http://scribu.net" rel="nofollow ugc">Author’s Site</a>

Filter Email Notifications

Stop WordPress from sending email notifications for comments that have been manually approved.

Scott Reilly

This plugin adds a checkbox to the profile page for users which allows them to opt into receiving a notification email whenever a comment is made to the site.
If a comment goes into moderation, only users who have the ability to manage comments on the site will receive the moderation notification email.
By default, all users of the site have the ability to subscribe to notifications about comments. A filter is provided to facilitate use of code to customize the feature’s availability to users.
Note: a “user” is a person with an actual login account for the site. The plugin does not facilitate permitting visitors who do not have an account on the site to be able to subscribe to all comments.
Links: <a href="http://coffee2code.com/wp-plugins/optin-comment-notifications/" rel="nofollow ugc">Plugin Homepage</a> | <a href="https://wordpress.org/plugins/optin-comment-notifications/" rel="ugc">Plugin Directory Page</a> | <a href="https://github.com/coffee2code/optin-comment-notifications/" rel="nofollow ugc">GitHub</a> | <a href="http://coffee2code.com" rel="nofollow ugc">Author Homepage</a>

Optin Comment Notifications

Checkboxes on user profiles to receive emails about new comments. Like it says.

Exactly what you want

Allows users to opt into receiving a notification email whenever a comment is made to the site.

Filter Email Notifications vs Optin Comment Notifications