The distinction is mainly a modern one to distinguish between two different styles. The Morion refers to the type with a down-turned brim while the Cabasset has a small, stylized peak at the top. Later in the 16th century you also sometimes see Morion-Cabassets which include elements of both.

At the time both would have been called a morion, chapeau de fer, or later a pikeman's pot. Sometimes period writers would distinguish between the "combed morion" and the "Spanish morion" but both seem to have been pretty widely used. The principles behind both the morion and cabasset are pretty similar. It provides good visibility and hearing, the comb/peak helps protect against blows coming directly from above, and the brim allows the wearer to protect his face from arrows or other projectiles by tilting his head forward.