When setting an iPhone up as a new iPhone, you aren't transferring a backup to the iPhone.
All iTunes content that was on your 4S should be in your iTunes library on your computer - music, tones, audio and print books, moves, tv shows, and 3rd party apps. You can transfer your chosen iTunes content in your iTunes library on your computer to the iPhone 4 via the iTunes sync process - make your selections under the various tabs for the iPhone 4 sync preferences with iTunes.
If the backup for your iPhone 4S was stored on your computer's hard drive which was updated as the first step during the iTunes sync process and you want to transfer the backup for the 4S to the 4, restore the iPhone 4 with iTunes and this time when prompted, select the backup for the iPhone 4S to be transferred to the 4 instead of selecting to set up the iPhone 4 as a new iPhone.