Can the content of an iPhone backup be extracted? After upgrading my device from iOS 4.3.3 to iOS 6, iTunes doesn't allows me to restore from an old backup, saying that it is an incompatible backup.

Why is this? Is there any way I could restore from this old backup? And, if not, is there any way I could retrieve the data from this backup?
There are several apps for extracting backups.
The one I have used is Iphone/Ipod Touch Backup Extractor, it's been around for a while now and it is freeware, just make a donation if you find it useful to help the developer keep it updated.

You simply select from which backup you want to restore by pressing the button "Read Backups", then select from the list of application settings that can be restored.
There is a detailed step by step guide here