App List Backup Pro

Saves the list of installed apps so you can re-download them from Google Play Store later (on the same or another device). This way apps still show up in Downloads and receive updates (unlike with other backup apps like MyBackup Pro).

The idea is NOT to back up non-Play-Store apps or to save you from re-downloading apps, so please don't rate negatively due to that.
The app saves a backup of your application list (or the part of it that you choose) onto the device's SD card or other storage. If both the old and new devices have an SD card slot, simply move the SD card from the old to the new device, and restore your apps. If one or both devices don't have an SD card slot, connect the old device to a computer, copy the folder called AppListBackup (from the root of the device's storage) to your computer. Then copy that same folder to the root of the new device's storage and run App List Backup to restore your apps.

This is the paid version of the app; it doesn't have any ads and does NOT require Internet permissions. There's also a free version available that has ads in it, but other than that is the same exact app. If you don't want to allow ANY permissions at all, including the "Storage" permission, use App List Backup No-Permissions, which uses email to store the backup and does not access your phone's storage.