Background: what (general) information should the panel know to properly judge the project. For example: What has been done before? How does your project relate to this previous work? If your project is about disease X, what is this disease exactly?
Significance is another word for "importance" or "impact". Why is the project important, for example: How many people are affected by disease X, and what do you expect to achieve when the project is completed. Better knowledge about the disease mechanism? A cure? A better detection mechanism? The structure of a protein involved in the disease (how does it help if we know the structure)? What they want to know is: How important is your project and how does it help to make the world better?
There is indeed some overlap between the two, but the requirement to write a section about the significance indicates that they want it to be separated from the background, without overlap.