Yes, there is a difference between brown paper and brown packing paper, even if both are made of kraft paper bags and look the same.
Brown paper bag:
Usually used to carry grocery items, lunch or retail items. It is pre-cut, east-shaped, and often has a flat floor. These bags are light and are designed for facilitating handling and carrying.
Brown Packing Paper:
It is a constant sheet or roll of craft paper that is mainly used to wrapping, cushioning or Voids filling in a shipping box. It is more versatile for packaging and provides better protection for objects during transit.