The reasons for this may vary somewhat with the specific circumstances of the country, but there are some general factors that apply in most cases.
Breaking away from the mother country is good in that it allows the country to create its own separate identity. By creating a republic, the country shows that it is run by its own people, not by some foreigner. On the other hand, the constitutional monarchy has the advantage of tradition. It helps the people in the country connect to their past and their heritage. It reminds them of where they came from without actually being any sort of a true restriction on their ability to rule themselves.