On the documentation page about user roles, it says:
Allows users to moderate comments from the Comments SubPanel (although a user needs the edit_posts Capability in order to access this)
For moderating comments placed on pages however (that have comments enabled) you also need to have the edit_pages capability it seems.
Notice that this will only work on posts and pages that the user him/herself created. If you want to give a user the capability to moderate ALL comments, you also need to assign them edit_others_posts/pages!
I have given the author of a site I made the following capabilities through the "members 0.1.1" plugin:
He can now moderate all comments placed on posts and pages that he created (or that I assigned to him).
As he will be the only person placing new posts on the blog, he won't need to moderate comments on posts made by others.
The good thing is, that I've made several pages with information, that should remain static. So I assigned those to myself (the admin), so the author cannot change or accidentally delete them. Comments are not allowed on these pages either.
I did make a "guestbook" page, with comments on obviously, but I assigned this page to be made by him, so he can moderate all comments in his guestbook.
Just wanted to share that this wouldn't have worked if you only assign edit_posts to someone who must have moderate_comments capabilities.
Took some trial and error, so I thought I'd spare others the time and worries. Once you understand how these capabilities work, it's pretty straightforward.