I finally managed to solve the problem by myself.
For those who face the same issue than I do, you will need to put your device into DFU mode, and proceed to a fresh install (no backups !).
After investigating a little bit on this issue, it seems that the device has a push notification ID : it remained the same even after restoring with backup. As it received double notifications with this ID, it apparently needed to be reset by a fresh install.
Actually, in my case, it seems like the problem came from the backup itself.
I first tried a clean install after putting my device into DFU mode, and then restored with a backup : it did not work, even with a backup created before the WWDC keynote event...
So I tried again, but this time setting it as a new device, and it now works just like before.
I hope this helps.