What are the differences between candy thermometers and meat thermometers?

I think Doug and yossarian both touched on the main points, but to summarize, there are four reasons why you might not want to use a meat thermometer for candy:

1. Range A meat thermometer might go from 140 F to 220 F or something like that, which is plenty for meat. Candy often requires a range from about 75 degrees (chocolate) up to 400+ degrees (hard candy). The range on most thermometers is simply not sufficient for most candies.
2. Accuracy When you temper chocolate, 88 F degrees is an ideal working temperature (for dark chocolate). Some people would consider 91 so high you might want to think about starting over. When you make caramels, the difference between 235 F and 240 F can be the difference between wonderfully chewy caramels and a sugar-flavored rock. If you can't read one degree increments at a glance, you need a real candy thermometer.
3. Speed Meat thermometers often take 30 seconds to a minute to get an accurate result. When you are making candies, you have to be able to tell what the temperature is right now.
4. Contamination I don't want the thermometer that goes in semi-raw meat anywhere near my chocolate.

If your meat thermometer is accurate and responsive you will be fine using it. However, some meat thermometers are slow and basic and not particularly accurate with no means of adjusting. With meat, once you are in a range of temperatures, you will have edible food, although it might not be exactly what you are looking for. With candy, a difference of a few degrees means a different kind of candy...again, still edible, but much more different than say between medium-rare and medium on your roast.

The goal with any thermometer is to have it be accurate within the range that you need to measure. Just because the range you want is on that thermometer doesn't mean that it is accurate in that range. If you trust it, fine. Worst case, you lose a batch of sugar water.

I have a digital meat thermometer which reads just about instantly, is accurate, and goes up to about 500F degrees. I can absolutely not see why I shouldn't be able to use it.

And I have no qualms in using a thermometer that has been in contact with meat. If you don't clean the tip after that, you have no place in a kitchen anyway.

If the temperatures you are looking for are within the stated range of the meat thermometer you will be fine. The difference between the two is just as you guessed, they are focused on different temperature ranges. A candy thermometer will need to be able to accurately measure melted sugar, which is way hotter than a medium rare steak. Putting a meat thermometer in to 350F sugar may break the thermometer. But putting it in to 160F fudge would be just fine.

And it's actually the pixies that will destroy your candy! Everyone knows there's no such thing as leprechauns.

Candy thermometer

Candy thermometers are, as you may have guessed, mostly used to measure temperature of a sugar solution whilst cooking. As sugar is heated it goes through stages and a candy thermometer can help you identify which stage of the process is happening, based upon the temperature.

It is well known that when you are making candy, you must stay within a certain temperature range. A candy thermometer helps you do just that, especially if this is your first time making some. The typical temperature range is from 220 degree F to 360 degree F. A quality candy thermometer comes with different ranges for specific stages of the candy making like soft crack and hard crack stage or the soft ball and hard ball stages. This is a scientifically based calculation and meaning you must pay attention when preparing candy because you can easily over- or under-heat the candy. This will destroy the whole recipe and wash your effort and expenses down the drain.

A meat thermometer is a thermometer used to measure the internal temperature of meat, especially roasts and steaks, and other cooked foods. The degree of "doneness" of meat or bread correlates closely with the internal temperature, so that a thermometer reading indicates when it is cooked as desired. When cooking, food should always be cooked so that the interior reaches a temperature sufficient, that in the case of meat is enough to kill pathogens that may cause foodborne illness or, in the case of bread, that is done baking the thermometer helps to ensure this.