The complexity of checking whether two CNF have the same number of solutions
The problem is coNP-hard; you can easily reduce the UNSAT problem to this problem.

A more precise characterization is that the problem is C=P-complete.  In fact, one definition of the class C=P is that it is the class of problems which are polynomial-time many-one reducible to this very problem (usually this definition is stated in terms of GapP functions).  But since this does not tell much, let me define this class in another way.

Let C=P be the class of problems which are polynomial-time many-one reducible to the following problem: given a boolean circuit φ and an integer K (in binary), decide whether the number of satisfying assignments of φ is equal to K.  By a standard reduction which shows the #P-completeness of #3SAT, we can restrict φ to be a 3CNF formula without affecting the class.  The class C=P contains a class called US, which contains both UP and coNP.

With this definition, your problem is C=P-complete.  Actually, the C=P-hardness is easy to see from the definition of the class C=P (which uses 3CNF formulas).

To prove the membership in C=P, suppose that we are to decide whether two given CNF formulas φ1 and φ2 have the same number of satisfying assignments or not.  Without loss of generality we can assume that the two formulas have the same number of variables, say n.  Construct a boolean circuit φ which takes n+1 bits as input so that the number of satisfying assignments of φ is equal to c1 + (2n − c2), where c1 and c2 be the numbers of satisfying assignments of φ1 and φ2, respectively.  Then the number of satisfying assignments of φ is equal to 2n if and only if c1=c2.