It doesn't matter. It is commonly used "the revolutionary war" but either way is fine.
American Revolution.
The Revolutionary War, also known as the American Revolution (1775-1783)
Revolutionary War
the American revolution is important because it led to the revolutionary war
revolutionary war (American Revolution).
The American Revolution
The American Revolution or the Revolutionary War.
Both of these are the American Revolutionary War. There was only one.
American Revolution
The American Revolution ended in 1783.
Who were the Continentals during the Revolutionary War??
The American Revolution a.k.a the Revolutionary War