Question

What is the difference between gauss theorem and gauss law ?

Answer 1

Actually both are same.
In certain matters, such as here, both the terms "law" and "theorem' are used. Otherwise normally "Law" is used for certain relationships which are found to be true on the basis of performed experiments. Examples are: Newton's Laws of motion and gravitation, Ohm's law, Coulomb's law. Theorem is normally preserved for logically proved assertions on the basis of assumed axioms which may have or may not have experimental verification. The Gauss' theorem which can be derived on the basis of Coulomb's Law. Now Coulomb's law is experimentally verified also the conclusions drawn assuming Gauss theorem are also experimentally verified. So there is no harm in calling it a law also. On the other hand the expression for electric field derivable from Coulomb's law due to point charge can also be derived, assuming Gauss law/theorem as an axiom. So some one can say that let us call Coulomb's law as Coulomb's theorem. After all the concept of a point charge is purely theoretical and one cannot perform experiments with it to verify coulombs law truly. Moral of the story is there is no difference between Gauss' law and Gauss' theorem. Its logical validity and experimental applicability is not affected by what name you call it..
Hope you understand!