After restore: no backup possible - iCloud "full"
I have an iCloud account with 25GB space, from which 8,8GB are free.
I have an iPhone 6, that has been constantly backed up to this account, including photos and videos (which take up about 13GB).
Now, today, my iPhone was exchanged at the Genius Bar. Of course, I have done both full local and iCloud backups, before, to have identical backups.
I have restored my new iPhone from the local backup (since I wanted to have a full restore including settings, passwords, account data, photos and apps, so I used the password-protected local backup from iTunes).
Now, iCloud tells me, it cannot backup this new iPhone to iCloud, since there is not enough storage space left. It says, it needs about 14GB of space, which exceeds the remaining 8.8GB of free space on the iCloud account.
I just don't get this: from the last iCloud backup with the previous iPhone6, nothing has changed, no new photos, not new data, nothing. All data stored in the cloud is exactly the data, that has been restored to the new, exchanged iPhone.
So please, any help is most appreciated: why does iCloud not "understand", that there is no new data?
Then, in my new, exchanded iPhone, I went to iCloud settings and deleted the old backup (from the previous iPhone) in the hope, that this would free the iCloud-space and I could just backup the new iPhone to the iCloud, again. But even that did not help, culprit seem to be the large "Foto-Mediathek" (German expression, don't know the english term) of 13GB, which I cannot delete nor make iCloud understand, that on the new iPhone, this data is exactly the same data as before....
I hope, I could make myself understandable and thank you for any suggestion or response!
iPhone 6, iOS 8.2