What is the difference between an apex and a vertex?

Well, at least in Polyhedrons, a Vertex is the point at the tip of a pyramid, apex is at the end of the cone.

An apex is the top of a pyramid or cone. Technically, the apex is a vertex due to its vertex-like corner. Prisms DO NOT have apexes. They only have vertexes because they do not rise up like a pyramid.

----------------------------------------------------------------

An apex is usually pointed end of an object; the tip; a point of culmination.

A vertex is usually the highest point or in mathematics , the point at which the sides of an angle intersect; the point on a triangle or pyramid opposite to and farthest away from its base.

Some apexes may also be vertices.