problems with user rights

I use two types of users. Normal users, and users with special rights. Normal users have access to their own profile and can edit data. Users they belong to both groups can't. I think there is a conflict of the rights and I have no Idea where I can change rights so that these users can edit their profile again.

What do you mean by they can't edit their profile? They can't reach profile edit page at all or can't see any tabs on it? If they can't see any tabs check the view access level you've assigned to various tabs within CB > Tab Management. If they can't reach it at all check the view access level you've assigned to your Joomla menu item to CBs profile edit page.

They can reach the profile and can change the tabs, but if they click save they get the message permission denied.

Please provide the exact message they're seeing. Check "Profile View Access Level" within CB > Configuration > User Profile and ensure it's accessible to the user (e.g. Public or Registered). It maybe saving their profile, but blocking access to their profile which it redirects to after a save. Aside from that the only other permission check applied is if they're trying to edit a profile other than their own, which isn't the case when editing their own profile.

They see:
Error

Access Denied
Edit Your Details

At CB > Configuration > User Profile is Profile View Access Level "Registered". Registered includes Manager, Registered and Super Users. Manager is my group with special rights.

There's no flat "Access Denied" error message in CB. If profile view or edit access is denied by CB you'll see a not authorized error. Access Denied is a Joomla error. Probably JGLOBAL_AUTH_ACCESS_DENIED to be more specific and is likely due to Joomla menu access permission. Ensure you've a public and published Joomla menu item to CBs profile page. If SEF is enable turn it off then cause the error and check Itemid in the URL and match it to its Joomla menu item then adjust the access for that menu item.