In A Nutshell
Comets are made of ice and rock, while asteroids contain no ice and can even contain metal. Smaller asteroids are called meteoroids, which are then called something different when they enter our atmosphere (depending on how it happens).
The Whole Bushel
There’s a lot of stuff floating around in the vastness of space, and one may wonder what the differences are—it turns out it’s not all just space rocks (although there are plenty of those).
Asteroids are generally pretty large—not just “size of a house” large (which they can certainly be), but “football stadium” large, several miles across in some cases. Smaller ones can be only a few feet across. Their most important feature, though, is their composition: minerals and metals, and notably, no ice. Comets, on the other hand, do contain ice as well as minerals, but generally no metals. Comets are also distinguished by their coma (tails) as they approach the heat sources of stars and the ice burns off into vapor; light from the star then illuminates the vapor, producing the tail. Both asteroids and comets usually follow fairly rigid orbits, as seen in the clockwork-like recurrences of famous bodies like Halley’s Comet and Hale-Bopp.
Asteroids smaller than a few feet across are called meteoroids. When one enters Earth’s atmosphere, it either burns up completely, leaving a trail across the sky (and is then called a meteor), or impacts the Earth’s surface (and is then called a meteorite).
Many scientists think that the Earth started out as a featureless ball of rock, and that most if not all of the metals and water on our planet were deposited here by asteroid and comet collisions. If so, then it looks like life on our planet was made possible by the very same types of collisions that some fear will eventually wipe us out.