Difference between sweating and sautéeing?

The answer is a bit complicated, because there is a confusing language issue here.

In standard cooking terminology, there is nothing in common between the two (except that both are stovetop). Sautéeing requires a wicked hot pan, a layer of oil (you can't use nonstick at these temperatures), and constant movement of the food. Basically, you are burning/caramelizing the outermost layer of your small pieces of vegetable while keeping the inside juicy. And if you'd let the food rest for a few seconds in contact with the hot pan, it will burn, so you have to keep it jumping.

Sweating is the process of allowing heat to slowly break down the cell walls of the plant. It is done on medium low temperature, and frequent stirring is counterproductive. In fact, if the pan is not crowded, you can do the whole process without stirring at all. It is usually done with fat, because this produces some very tasty byproducts (I've read of a study found that one of the flavors people found to be most pleasant in meat was in fact produced by a chemical reaction of onions and butter). But the fat is not strictly necessary for the process to take place. The resulting vegetables are soft through and through, and have lost some of their water during the cell wall breakdown. It needs some experience to find the temperature at which the food won't burn, but the juice will cook off quickly enough to not turn the bottom layer limp.

But this terminology is not common among home cooks at all in English speaking countries. I have never seen the term "sweating" in an English recipe meant for home cooks. And for some reason, they use the term "sautéeing" instead, even when they are clearly not sautéeing at all in the culinary school sense. I don't know what has caused this strange language phenomenon, but it has led to a somewhat confusing situation.

The result is that you have to know your source. If it was written for and/or by home cooks, you can assume that there is no difference at all. When you read "sautéeing", you should sweat your vegetables, or brown the meat. If you are using a professional cooking resource, you need to use the other definition. Also, if it's for home cooks but a translation from a language which makes the distinction (e.g. German), it is possible that it uses culinary school terminology. It's a bit like the dinosaur/bird terminology problem in biology: depending on who is talking, it means different things.