Actually, there is no chemical difference between hair and fur. The difference is merely a nomenclature. We call this keratin protein outgrowth as hair when it is on a human, we call it fur when it is on a non-mammal. We call it whisker if it is around a non-mammal's mouth, we call it mustache if it is on a human's mouth. Etc. Okay, English is only my 3rd language, so I'm not very good with nomenclature. It took some study for me before I remember to call it mustache instead of beard. LOL. But yeah, a lot of people call it hair instead of fur because unlike bulldogs for example, bichon hair keeps on growing. Technically, it stops growing... it's just that the "stoppage point" of bichon hair is a lot longer than other dogs and the show "look" of the bichon is shorter than the "stoppage point" (as opposed to say, the pekingese who sports his full length outer coat for show).
Bichons still do shed. But, because of the way the fur grows on a bichon, the hair that is shed gets trapped instead of dropped. A bichon grows "guard hairs" around the time they turn 1 year old (also called a double-coat). The guard hairs are coarse fur that grows over the soft fur on a bichon's back which causes the hair to curl and act as a trap for loose hair. This is why bichons have to be brushed everyday because the loose hair will end up getting matted into the coarse hair which can cause discomfort, pain, and in extreme cases, hematoma. And bichons are not really "hypoallegenic" either. It's just that the dander gets trapped on the guard hair as well. And because most people brush their bichons regularly, the saliva deposit on the hair gets brushed out with it. But, a person who is allergic to dogs would still have a reaction when doing the daily bichon brushing maintenance. So, I always tell people who have allergies who wants to have a bichon for its "hypoallergenic" qualities - unless you can assign somebody else in your house to brush your dog's hair daily, don't get a bichon.
And this is probably a lot more info on the bichon than anybody cares for. LOL. I love my bichon just as much as I love my bulldogs.