RS codes are a sub-class of BCH codes, where the subfield and field are the same.

RS codes, where they exist, achieve greater distance than BCH codes... but they have a far more restricted set of parameters than BCH codes.

Many applications are moving to things other than BCH codes because even though they are a much better class than RS codes they are still limited and the existing BCH codes at many sizes are not as good as other kinds of codes.