Billed weight is the weight calculated by the carrier based on the either the size of the box or the actual weight (whichever is greater). Billed weight gets calculated by the carriers on every shipment that gets processed in the system by default, though the calculation is only as accurate as the data (i.e. if dimensions aren’t entered by the user, only the actual weight can be taken into consideration).
Here is a link describing the difference in more detail: