As a noun, capital refers to (1) a city that serves as a capital of government, (2) wealth in the form of money or property, and (3) a capital letter. As an adjective, it means (1) principal, (2) involving financial assets, and (3) deserving of the death penalty. There are other definitions of capital, but these are the main ones.
Capitol has two very specific definitions (outside Ancient Rome): (1) a U.S. state legislature building, and (2) the U.S. Capitol building in Washington, D.C. State capitols are located in the capital cities of U.S. states, and the Capitol is located in the capital city of the U.S. The U.S. Capitol is capitalized. State capitol buildings are not.
Examples
These writers use capital well:
The ability to expand depends on access to capital … [Capital Spectator]
Endeavor With a Capital E [Collecting Raindrops (now offline)]
In 1972, the U.S. Supreme Court banned capital punishment, but the decision was overturned in 1976 … [Columbia Missourian]
You are a capital fellow, Planchet … [Ten Years Later, Alexandre Dumas]
These writers use the narrower and less useful capitol correctly:
The House will take up a bill this week adding one statue of a District luminary to the halls of the Capitol … [Washington Post]
Cherry has spent 28 years at the Capitol as a lawmaker and lieutenant governor, and at one point, he hoped to be Michigan’s next chief executive. [AP (article now offline)]
