Supercomputers are the biggest, fastest, and most expensive computers on earth. The cheapest supercomputer costs well over $1 million. They're used for scientific simulations and research such as weather forecasting, meteorology, nuclear energy research, physics and chemistry, as well as for extremely complex animated graphics.
Besides raw speed, one big difference between a supercomputer and a mainframe is that a mainframe serves many people at once or runs several programs concurrently, whereas a supercomputer funnels its power into executing a few programs at high speeds. Mainframes are mostly used for large data storage and manipulation tasks, not for computationally-intensive tasks.