Beginner Tesla Question

I just became interested in Tesla Coils a little while ago. I've seen 
pictures at pupman-dot-com, and other websites. I also saw a picture of a Tesla 
Coil in Edmund Scientific. In Edmund Scientific, it looks like an upside-down 
ice cream cone of coiled wire. At pupman-dot-com, they look almost identical to a 
Van de Graf Generator. What is the difference between the two, and how are 
they used differently?