What phone did you move from, Android, iOS? Backup Assistant is for your contact list and doesn't backup photos, texts, and etc.
Verizon has a very good guide on how to save and move content from one device to another. Microsoft has a migration tool also that helps when moving from a non-Windows phone.
You'll probably need to copy your photos, video, music, and etc. from your old device to a computer or the cloud and then move them from there to your Lumia Icon.