You are correct that under classical assumptions of logic ("normal" mathematics), the Dedekind Cuts and Cauchy Sequences indeed result in the construction of isomorphic objects.
However, under constructive assumptions of the logical axioms of mathematics, such as intuitionism (not accepting the law of excluded middle, i.e., assuming that the statement "P and NOT P" is not necessarily true for all P), the construction of the Dedekind Cuts and the Cauchy Sequences may result in different objects, i.e., it may no longer provable that they are equivalent.
You are correct that under classical assumptions of logic ("normal" mathematics), the Dedekind Cuts and Cauchy Sequences indeed result in the construction of isomorphic objects.
However, under constructive assumptions of the logical axioms of mathematics, such as intuitionism (not accepting the law of excluded middle, i.e., assuming that the statement "P and NOT P" is not necessarily true for all P), the construction of the Dedekind Cuts and the Cauchy Sequences may result in different objects, i.e., it may no longer provable that they are equivalent.