A commonwealth is a united group of individuals or states with common or shared objectives and interests (see Republic). An American state that chooses to refer to itself as a commonwealth rather than a state is making the overt statement that it views itself as autonomous, maintains its own authority, has chosen to participate in the union of states (known as the United States of America) of its own free will, has not given up its right to independence and self-governance, and has granted the federal government limited governing authority while maintaining it's own greater governing authority over itself. The U.S. was formed on the basis of severely limited federal governing authority and, in comparison, much greater state governing authority. The chosen title of commonwealth over state both emphasized and codified that fact and that recognized agreement.
The U.S. was designed to have a state heavy government vs. federal heavy government and some states chose to make a point of clarifying this by identifying themselves as commonwealths rather than states as the entered the union of states and became the Unuted States of America.