Backstory includes the events that directly lead to the "story" and its storyform. For example:

Bob's backstory involved the loss of a child pet that made him hypersensitive to having children, pets, or any other form of personal responsibility. The story begins with the death of the Bob's only living relative and his custody of Aunt Betty's fifteen year old cat, Wobbles.

Background, for a character, includes the elements of his or her history and experience. Things such as family history, education, work experience, hobbies, etc. can fit into a character's background. These may, or may not, be relevant to the character's involvement in the story, but do help to describe how a character might be prepared for varying scenarios. For example:

Rita's background includes being valedictorian of her class, graduating suma cum laude, and having a degree in Russian literature with a minor in Restaurant Management. She's an Army brat and lived in thirteen different countries while growing up with her Siamese twin brothers. None of this prepared her for working at the Taco Mart.

So, the main difference between Backstory and Background is that Backstory is directly tied to the essential growth and development of the story's storyform, while Background provides an environment within which the characters have individual, historical contexts as an aspect of the story's storytelling.