Historically, water was hardened to raise the pH to reduce the leaching of lead from lead pipes, which could cause lead poisoning.
The advantage of drinking hard water is that it will supply calcium in the form of calcium carbonate (chalk), which can reduce the risk of osteoporosis. In order to be able to absorb it, however, you will need to have a good supply of natural vitamin D in your diet. Calcium carbonate is also poorly absorbed compared to dietary calcium that is found in, for instance, dairy products and nuts.
The disadvantage of hard water is that it tends to clog up your kettle and your washing machine, and it has (in my opinion) a sour taste. It also leaves a scum around the cup when you make tea, and gives tea an off taste.
Hard water also tends to be treated with chlorine, which is used to kill off harmful pathogens such as typhoid and cholera.
Chlorine also kills off the friendly bacteria in your gut, which are essential for good health, so if your local mains water is chlorinated it is recommended that you use a water filter containing activated carbon to remove the chlorine.
Fluoride is sometimes added to public water supplies as a cheap way of disposing of poisonous fluoride emissions from the aluminium and fertiliser industries.
Even moderate levels of fluoride can cause dental fluorosis, where the calcium structure in tooth enamel is affected by fluorine, causing a mottled, glowing effect. Contrary to protecting teeth, the porosity of the sub-surface enamel is increased by fluoride, leading to more cavities in adulthood.
Fluoride has been promoted for reducing cavities, as statistically there are less visible cavities in children in areas with fluoridated water. However, fluoride only reduces the visible appearance of cavities and over time the decay continues inside the tooth, due to the increased porosity of the sub-surface enamel. When the cavity is finally visible, much more extensive work is needed requiring an unnecessarily disproportionate use of filling material (and if amalgam is used, unnecessary exposure to mercury).
Also, a linear correlation between the Dean index of dental fluorosis and the frequency of bone fractures is observed among both children and adults.
(Alarcon-Herrera MT, et al. (2001). Well Water Fluoride, Dental fluorosis, Bone Fractures in the Guadiana Valley of Mexico. Fluoride 34(2): 139-149.)
Fluoride is actually poisonous (as is chlorine), and one tube of fluoridated children's toothpaste contains enough fluoride to kill a child.
Too much fluoride can cause osteosarcoma (a very rare bone cancer), reduced intelligence in children, and impaired thyroid function.
Fluoride exposure has been blamed for Crohn's disease, IBS, joint pain and other auto-immune disorders, and kidney and liver damage.
Fluoride at levels as low as 1 part per million (ppm) has been shown to affect the ameloblasts responsible for secreting enamel in teeth.
Studies have also shown that fluoride can disrupt the function of the pineal gland - "a small gland located between the two hemispheres of the brain that regulates the production of the hormone melatonin. Melatonin is a hormone that helps regulate the onset of puberty and helps protect the body from cell damage caused by free radicals.It is now known - thanks to the meticulous research of Dr. Jennifer Luke from the University of Surrey in England - that the pineal gland is the primary target of fluoride accumulation within the body."
Therefore, if your local water supply is fluoridated, you should filter it before drinking, and you should change the filter regularly. You should also campaign to stop it being added at the source.
On the other hand, soft water (and filtered water), does not clog up your kettle or washing machine, and it doesn't make a scum on your tea.
However, soft water can still contain chlorine and fluoride and it is still recommended that you filter it to remove these poisons.