How can I restore photos from an iCloud backup to a new iPhone?
Photo Stream and iCloud Backup are separate — your photos are stored in your iCloud Backup and many of them are duplicated and stored in Photo Stream (unless you deleted them from your camera roll in which case they're not duplicates).

The iCloud backup can be restored whole to your device. You can't restore parts of it — it's all or nothing.

However, there is a workaround:


Make a backup of your new device to iTunes.
Reset your device to factory settings and restore the iCloud backup.
Use Image Capture or iPhoto/Aperture to import the photos.
Reset your device to factory settings and restore the iTunes backup.
Copy the photos back to your device.