I promised mutations, and mutations ye shall receive. For any of these mutations that cause hairlessness, the rats are fertile, but they all tend to have problems with raising the babies to weaning age. All these 'hairless' rats have whiskers, and some hair on the body, but it is short and sparse, and cyclic growth, loss, and regrowth can occur in any of these rats. The only real way to tell what type of hairless you have (if you have them) is through blood work. The nude rat has no thymus, so will have low or no T lymphocytes. The fuzzy rat will have signs of kidney disease.

Nude locus... Rowlett nude (rnu) - recessive on Chromosome 10
Rats that have a mutation in the nude locus have sparse hair on the face and head, but not really anywhere else on the body. Before these nude rats are weaned, they may also have a flakiness to their skin, but it does later become smooth. Nude rats may also grow hair during their lifetime, but will lose it again shortly. So, what exactly is this mutation and what does it do beyond the physical loss of hair? The nude locus codes for a protein that is required for normal development of skin, hair, and thymus, so the mutation effects all three. Nude rats have no functioning thymus, which leads to immune deficiency and a smaller stature, on top of their hairlessness and possible skin abnormalities. These rats rarely make it past 9-10 months old.

Fuzzy locus... (fz) - recessive on Chromosome 1
Fuzzy rats have short, rough hair as youngsters, but by around 2 months of age, the coat is either very sparse or completely hairless. By 6 months, they are usually smooth-skinned and will stay that way through adulthood. If it can be said that one of the hairless mutations is worse than another, this one would be it, with the highest rate of physical abnormalities. The most prevalent and most deadly is kidney failure. Fuzzy rats suffer from chronic kidney disease, and ultimately, this is what causes the early death in these rats. Most only live to 16-20 months of age. Tumor incidence is also affected in fuzzy rats and, while they do not have a higher risk of them, they do develop them earlier in life than furred rats. Adrenal glands also tend to be cystic and dental malocclusion is extremely common, to the extent that more than 50-75% of fuzzy rats will need regular tooth trims to eat and survive somewhat normally. Of course, the skin and hair is affected as well. Cysts can form in the skin because the skin tends to be more thickened, and it is this thickening that is the reason for the hairlessness (and cysts) in these rats. The hair cannot penetrate the thickened skin, so the follicle can become inflamed. One of the mutations of the fuzzy allele is the Charles River hairless, but it should not be confused with any other mutation. It is still the fuzzy mutation that causes this type of hairless rat. In fact, this is where there is some confusion. There is NO true hairless mutation in the rat, as there is in the mouse. The only currently known mutations in rats that cause hairlessness are the three discussed here.

Shorn locus... (shn) - recessive on Chromosome 7
Shorn rats have a patchy, thin coat as adults, usually with a mask of hair on the face, and more ragged spots of short hair on the rest of their body. As weaning babies, they usually have short hair on the body, but not on the face, but by around 8 weeks they are usually hairless and the facial pattern of fuzz gets the mask like look of the adults. This mutation not only effects hair, but also effects kidney function, and the mutation creates an error in the coding for keratin, which is why the hair loss occurs. Most shorn rats have abnormalities in the kidneys and heart, are smaller than normal rats, and tend to die at a younger age averaging 10-14 months old. The shorn rat is the least likely of the three to be found in the pet trade or rat fancy.

References: 
http://sdb.bio.purdue.edu
http://www.biology.ccsu.edu
http://www.spflrc.org/user/rats/CoatTypes.htm 
http://www.rmca.org/Articles/truehairless.htm