No, they are two entirely different phenomena.
A planetary nebula is born when a low mass star (like the Sun) dies, while the supernova is the death of a massive star.
In low mass stars, the outer envelope of the star at the red giant stage is ejected out into space, while the core of the star becomes a white dwarf. The ejected envelope forms a planetary nebula that expands away from the central star.
In high mass stars, there is a rapid compression of the star followed by a fast ejection of the compressed star material. The end result is a cataclysmic explosion called the supernova, leaving behind a neutron star or a black hole. The exploded remains of the star form a supernova remnant.
The material released in a supernova is denser than in case of a planetary nebula. The planetary nebulae are made of a variety of elements, ranging from hydrogen, to helium, to carbon. Supernovae produce elements such as hydrogen and helium as well as heavier elements (heavier than iron).