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.
The similarities and differences between state government and a regime are varied and subtle.
When a large territory is united under one government, yet has divisions of smaller territories within itself, you have the setting for two sets of laws.
In a state, the rules of the law are dictated by the state. As long as they do not go against the rules of the country, they are the laws that are enforced. For example, a federal law may give a person the right to own crayons. The state may set up a controlling law which says that ownership is fine, but you may not use them in a state-owned establishment. The state accepts the federal law but limits it within its own borders.
In a regime, there are still the two sets of laws. The difference being that the larger territory defines the laws of the smaller one. The federal government says, “You may have crayons, and since you are a part of OUR country, you may use them wherever you please.” The smaller division of the country has no power to restrict crayons within areas particular to it.