I always felt as if the two approaches demonstarate a difference in philosophy as to how one might approach real analysis.
The cantor construction is more geometric, it works for any metric space, even being characterized by a univeraal property. The cantor construction solves the issue of having missing points in order to carry out geometric arguments, and is constructed essentially in a way that fills those holes and gives a "universal" geometric space over Q.
On the contrary, dedekind construction is far more analytical, it relies heavily on the ordered properties of the field and introduces the imporant analysis formalism of infimums and supremums that is crucial to give a well assgined numerical quality to many measurements like the integral and to allow onr to perform arguments like continous induction.
In my head, the cantor construction sees the reals as points on a line, whereas the dedekind construction sees them as numbers with value. And one of the beauties of real analysis is that both of those approaches go hand in hand and lead to the right things we want.