What is the difference between a Liberty Silver dollar and a Peace silver dollar?

The biggest difference is Peace dollars exist and Liberty dollars don't. A very common mistake non-collectors or amateurs make with US silver dollars is they think because the Word LIBERTY is on the obverse of a Peace dollar (or other coins) it's called a liberty dollar. No US coin is referred to as just a "Liberty" coin. Also this comes up with Morgan dollars, but with the national motto E PLURIBUS UNUM because it's on the front of the coin.