When you march on the state capitol to protest a bill before the legislature, you are assembling outside a building that houses the state government.
Use the noun capitol when you're talking about the building where a legislature assembles to govern a state or region. In the United States, each state has an individual capitol building, and the federal government has one too — although when you mention "the Capitol" in Washington, D.C., it is capitalized. Speaking of which, be careful not to confuse capitol with its homophone, capital. Capitol comes from the Latin Capitolium, a famous ancient Roman temple.