Kummer's theory describes abelian extensions of arbitrary fields containing roots of unity, in a uniform way over all fields (i.e., the answer doesn't depend on the dimension of the field and all such.) Class field theory describes abelian extensions of number fields, and apparently it can be extended to fields of higher dimension, but the answer then will depend on the dimension of the field.

The answers given by Kummer and c.f.t. are related via Tate's duality in cohomology, which depends heavily on the dimension and other special properties of the field. E.g., the local and the global case are usually treated separately and the answers obtained for them in the c.f.t. are quite different. (Perhaps there exists a unified treatment of the local and global cases, but I am not aware of it right now.)