Backgrounds provide the extra mechanics you mention in the first sentance, but not the story part you mention in the second.
Your backstory is what actually defines in story termsp who your character was before they go out and adventure, and the backstory would do this whether or not they had a background.
Further backgrounds, as they are, restrict the backstory elements. What if my father weas an adventurer and I was born into the adventurer life (not a farmer, or a soldier, or a sailor but an adventurer)? What if my character was a serf on an elemental plane? What if I was born as part of an Orc war tribe?