Think of the budget calculation as the calculation of a surface integral. FEFLOW recalculates the fluid flow into and out from the model. The result is a boundary budget, which does not have any information of what happens inside the model. I attach a pdf document showing the background of the budget calculation in FEFLOW.
For getting differences in storage (by phreatic surface movement), you can use the content analyzer which does an integration of the water content in the model for the current time step. Comparing the water content in two time steps, you can get the storage. But: The content analyzer currently (in version 5.2) does not consider storage by compression (result is just water volume in m³).
Theoretically, i.e., assuming that the content and budget analyses are done without the slightest error, the difference between the boundary budget and the change in water content would be the model error. Practically it will be still a combination of the model error and little errors in summing the boundary flow at nodes, integrating the water content (especially for unsaturated models), etc.
