The Oxford English Dictionary defines a regime as "a government, especially an authoritarian one." It defines a state government as follows:
The government of a nation or state; now specifically (also with capital initial(s)) the government of one of the states of the United States of America, the Commonwealth of Australia, etc. (sometimes as distinct from the federal or national government).
The two terms do not align perfectly, so we need to look at what each one connotes or implies. The terms are alike in referring to the governance of a country or territory.
A regime, however, implies a country ruled by a top-down, tightly controlled central government. An authoritarian government is not usually democratic: it does not reflect the will of the governed but instead imposes its will on them, whether they like it or not.
In contrast, the existence of state governments implies a dispersal of power away from central authority and into local hands. The definition's examples of state governments in democratic nations, such as the US and Australia, suggest that state governance is a bottom-up, democratic affair in which the sources of power are local and closer to the people being ruled. While state governments certainly have the potential to be authoritarian, they nevertheless represent a dispersal of power that works against centralized authority.