Yes, but the new phone must be at the same version as the backup or newer. If the new device comes with an older version, set it up as new so you can get it to connect to Apple's servers and upgrade it to 8.4. Then wipe it clear (Settings / General / Reset / Erase All Contents and Settings) so it returns to the original state but with the new OS and then Restore from your backup.
What's saved in the backup depends on how and where you did the backup. I can state from having done it over 4 times already that a backup done using iTunes on a Mac, set to be stored locally on the hard drive with password, with apps/movies/music/photos/books to sync with the iTunes' Libraries will Restore onto a new phone leaving it identical to the old one. Have never done in any other variant (PC instead of Mac, iCloud, no password, etc.).