Tennessee is not a commonwealth. It is one of the 50 states of the United States, admitted June 1, 1796 (16th state).
Only four U.S. states use "Commonwealth" in their official name:
Legally: nothing. It's a stylistic choice from revolutionary-era republican ideals (common weal = public welfare). No constitutional, fiscal, or governance difference from "state." All 50 are equal under the U.S. Constitution.
Constitution of Tennessee (1796, current 1870): "The State of Tennessee." Statutes, seal, letterhead: State of Tennessee. Never "Commonwealth."