In strictly volcanic terms, they are two different kinds of depressions. A caldera is formed when a large eruption of magma, or lava, leaves a gigantic empty chamber underground. The volcanic material above the chamber, usually made of volcanic rock and tuff, collapses into the empty magma chamber.
Yellowstone National Park is mostly a giant caldera, as is the Valles Grande caldera in New Mexico and others--remains of giant eruptions in the past.
A crater, on the other hand, is almost always a vent for volcanic activity. Large volcanoes form when magma and other material is ejected from a vent, which builds up a cone around the vent, and as more material vents over time, a mountain is eventually formed.
Take Mt. St. Helens in Washington State, for example, the most recent eruption we have to study. The eruption literally blew away the top 1800 feet of the mountain, leaving a large crater. It is not the result of magma being displaced and then the land above collapsing in on itself, but rather, an explosive eruption. There is a large lava dome in the center of the crater as the vent there continues to do what it has always done, eject magma.