A great distance from the Sun, asteroids aren't much to look at; it lacks both luminance and a tail, and it's only permenant part is it's tdark, dusty nucleus (which, on it's own, is neither notable or unique). Think of a comet as this gigantic snowball with some dirt in it and you're almost there, however, the chemical composition of a comet varies slightly from a snowball found here on Earth - according to the information produced by the Giotto spacecraft as it flew by Halley's Comet in 1986. In addition to water ice and dust, it also contains ice made out of carbon dioxide, methane, ammonia, and other trace gases.
The reason why we can see comets at all is because those we can see have highly elliptical orbits. So, they spend most of their lives in darkness of space, and for a brief duration they sweep into the inner solar system for a close encounter with the Sun. When it reaches the vicinity of the Sun and it's radiant energy begins to bombard the comet, vaporized gases and jets of dust erupt from the cometary nucleus, forming parts of the comets we see from Earth - the coma and the tail.
A comet's coma is the concentrated gas and dust that forms the "head" of the comet, and it can be impressively large, up to 10 times the diameter of the Earth itself. The tail of a comet is produced once material originating from the cometary nucleus erupts outward, and then the solar wind pushes it away from the radiant source (star).
Asteroids are leftover fragments created during the initial formation of our solar system and are mainly stony objects, although some are a mixture of rock and metal, and some are almost mostly metal (these are the ones deep thinkers dream one day of mining for their resources).
Asteroids have been the center for doomsday scenarios that have been perpetuated by big Hollywood productions such as Deep Impact and Armageddon. Although it is unlikely that such a cataclysmic event is going to occur soon, scientists have raised concerns over the lack of an adequate global asteroid-detection system. Right now, if a car-sized asteroid were heading on a direct path for Earth, we probably would not be able to detect it. Depending on the asteroid's compositon, the subsequent impact could potentially cause great damage to property and a sizeable loss of life - especially in urban areas. If it were to hit the oceanic areas of our planet, it may create a destructive tsunami which could flood coastal cities in the vicinity.