In terms of constitutional rights, there is no real difference between a commonwealth and a state in the United States. “Commonwealth” is an old term that has largely been replaced with the title of “state.”
In the 17th century, the term “commonwealth” referred to an organized political community, or what today is called a “state.” Four states in the Union—Kentucky, Pennsylvania, Massachusetts and Virginia—are still officially called commonwealths. When they joined the United States, they chose to keep the old form instead of using “state” as their title.
In modern times, the term commonwealth also refers to “a political unit having local autonomy but voluntarily united with the U.S.” Examples of these types of commonwealths are Puerto Rico and the Northern Mariana Islands.