I own two Curlies and they are definitely a breed I have grown to love. Within the breed the horses can vary greatly. Quoting from the ABCR website; "Due to the years of out crossing, Curly Horses will be seen with different conformation, size and weight. This influence will probably continue for the next five to ten generations of Curly-to-Curly breeding.
You can find the [ABCR] breed standard here
They are pretty much like any other horse, just add curls! Every Curly I've met also seemed to be more personable than most horses. While my Appaloosa gelding loves people and will whinny at you when he sees you, Tenakee, one of my Curly mares, will always come up and greet me even if I'm on the other side of the pasture. Considering she is pregnant it is quite the waddle for her but she is committed to saying hello.
Their grooming isn't too difficult in comparison to a regular horse. That curly coat will really hold in dust and dirt, though. But a bonus is when you have a curly coated horse you don't notice where the saddle has been as easy. (Compared to a normal horse where the coat would be all messed up.) One thing is mane upkeep is killer, but crucial, assuming you get a Curly with a lot of mane. Keeping them braided makes the curls more uniform and easier to handle. But some Curlies shed their manes and tails all together so upkeep depends on the horse.
They can be used for anything, I've seen barrel racers and dressage stars, they are a very versatile breed. But seeing as they have different influences from all different breeds some are more acclimated to certain disciplines. As for their 'hypoallergenic' qualities I'm unsure. They are not truly 'hypoallergenic', but simply don't get as much of a reaction out of people allergic to horses as a regular horse would. Their are also straight-haired Curlies, who are a contradiction, but are said to also posses that same allergy relief.
If you have any more specific questions feel free to ask and I will try to answer best I can. Otherwise I can spout off twelve pages worth of nonsense on how great Curlies are. ;3