Problems Restoring to Backups

Question

Yesterday I went into an Apple Store for a warranty service on my iPhone 4S.  It was an older phone and the lock screen button had become stuck and unusable. Knowing that I might be getting a new phone I backed up my iPhone to both iCloud and iTunes. Because the phone was under warranty, Apple replaced the phone which fixed the hardware problem I was having.
I activated the new phone on my network and tried to restore it to my original settings when I got home. However after following the initial steps of the setup assistant I reached the point where I needed to chose what file to backup from. To my surprise there was no option for an iCloud backup at all (this is what I am more familiar with and planned to use). Because I also had a recent iTunes backup, I decided to proceed in the process with that file. When the phone finished restoring many of my apps were missing or not updated. My wifi was greyed out/disabled and the phone generally was nothing like the one I had backed up originally.
I don't know if I'm missing something basic, but my setup assistant does not have any iCloud restoration options. My computer registers a complete backup in iTunes under the Device Preferences page. I don't have any other recent iTunes backups. If someone could help me get my backups operating fully on the new phone that would be fantastic.
The Geniuses at the Apple Store made this sound easy and didn't offer any words of advice. Apple had kind of let me down on this so far so any help at all would be appreciated.

Answer 1

Backups are responsible for saving all user settings including apps stored on the phone and the data associated to those apps. ckuan is correct in saying iTunes or the App Store is responsible for pushing the apps back to the device when restoring. The apps should be greyed out when a restore first begins, be patient as it can take a while for them to install.
When you have greyed out wifi, that means your phone has lost track of the wifi antenna attached to the logic board. This could be due to your backup, meaning that it could be a corrupted backup, or it could be a hardware failure which sometimes happens when you bring a corrupted backup to a new device. I have had this happen to me several times on replacement devices in the years I have used since the iPhone 3G.
My recommendation would be to ensure your backup's timestamp in iTunes is the one you'd like. If that checks out I would recommend attempting another restore by erasing all content and settings on your device. Go to Settings>General>Reset>Erase All Content and Settings. After that attempt to go through the process of setting up your phone. This would allow you first to ensure the WiFi antenna is recognized by connecting to a WiFi network in range(preferably yours). Then I would recommend attempting an iCloud restore. Good luck.