As far as I know an anti-sweat sheet is actually supposed to absorb the moisture from the horse's coat whereas cooler is supposed to ensure the horse dries slower to prevent him from getting cold. So a cooler is something you put on a horse while walking him after workout while anti-sweat sheet you can put on him or under a regular blanket when putting him into the stall (and removing it after some time when it's wet).
Okay, I just googled it and found so many different opinions as people in different locations tend to call different things different names. Some websites say both are the same thing, some say the difference is that cooler is made for cold weather while anti-sweat is used in summer, some say the difference is in thickness and material. It would be nice if someone actually knew what was the original difference between those two before people started to mess with the names. Or maybe it really is just two names for the same thing...?