An Independent State is a:
nation, country, or state in which its residents and population, or
some portion thereof, exercise self-government, and usually
sovereignty, over its territory.
Whereas a Free State is:
occasionally used in the official titles of some states throughout the
world with varying meanings depending on the context. In principle,
the title asserts and emphasises a particular freedom of the state in
question, but this is not always reflected in practice. Some states
use the title to assert sovereignty or independence from foreign
domination, while others have used it to assert autonomy within a
larger nation-state. Sometimes "free state" is used as a synonym for
One can only surmise without access to the contemporaneous records of the Gouvernement de la République Française but it seems that the reference to Vietnam (or French Indochina) as being a Free State may have been way for France to announce to the world her military, political and/or commercial intentions in the area, and to give the Vietnamese people some degree of autonomy without France relinquishing overall control regained following the 1945 to 1946 War in Vietnam.
For context: the geopolitical events in and around Indochina at the time were complex with changing loyalties, fragile allegiances and contested borders with the main events being the Japanese invasion in 1940, the 1945 coup d'état, the Japanese and Vichy French surrender at the end of the second world war, and the the First Indochina War in 1946, but a more detailed answer may be available at HistorySE or PoliticsSE.