Are hard metal and heavy metal the same?

No.
In metalworking, "hard metal" is a commonly used name for cemented carbide, which is a sintered hard material based on wolfram (tungsten) carbide. Cemented carbide is often referred as simply tungsten carbide. It is the main cutting tool material used today.
Heavy metals are metals with high atomic weight or density. In the metalworking industry, the term “heavy metal” usually refers to heavy metal alloys, which are sintered composite materials containing 90% or more tungsten.