Generalized Weyl algebras, as defined by V. V. Bavula [St. Petersburgh Math. J., 1993], are a family of algebras containing both some classical objects (enveloping algebras and their prime quotients, Weyl algebras, invariant sub-algebras,...) and their quantum analogues. These algebras are generated by two generators over a k-algebra R, with relations given thanks to an automorphism and a central element of R. We are interested in problems of classification for such algebras. In this talk based on a joint work with L. Richard, we will consider isomorphisms between generalized Weyl algebras, giving a complete answer to this problem in the quantum case for R = k[h]. We will give separation results too up to rational equivalence and Morita-equivalence for these algebras.