The role needs "edit_post" capability to edit comments (the reason beeing that comments are posts for wordpress)
--> do your authors have "edit_post" capability?
this enables the role to edit posts also, which might not be desired.
Wordpress capabilites are weird.
There exists a meta capability 'edit_comment', but its not very intuitive. As far as I understand it currently, you can not directly assign it to a user (you can assign "edit_comment" capability, but it seems to be treated as custom created capability and therefore ignored..
I tried to let my subscribers edit their own comments, but
seems to have no effect for me, you can try it for authors but it probably won't do the trick.