Why do I need special shoes for dancing?
Sneakers or comfortable shoes don't typically work very well for ballroom dancing. Most partner dances have some pivoting. If your shoes are too sticky, at best it will put a strain on your knees and ankles, at worst you risk a sprain or other injury. But if your shoes are too slippery, you will have a tough time keeping your balance on a quick turn. That's why you might want special shoes. From the perspective of the dance venue, street shoes may leave scuff marks on the floor.
Ballroom dance shoes are specially designed for dancing. The shank provides an appropriate amount of support, and they are much more flexible than regular shoes. They have a suede sole (known as chrome leather), which gives just the right amount of traction to allow your feet to slide around a good hardwood floor without going out of control.
Ballroom dance shoes will cost in the neighborhood of $100 and up.
What shoes to wear where?
Unfortunately, chrome leather soles are rather sensitive to environmental stresses, like water, tar, oil etc. As such, their use is normally restricted to ballroom dance venues. For such dance spots as night clubs, cafeterias with linoleum or concrete floors, outdoor dance parties, etc., a pair of leather-soled dress shoes may be preferable.
Dress shoes with smooth leather soles can be adequate for dancing, depending on the floor and the shoes. Some shoes might need some scuffing up of the bottom to make them more non-slip. Dress shoes might also be somewhat stiff for dancing, and could result in sore feet after several hours. The advantage is that there is no need to change from street shoes to dance shoes and back.
Many manufactures make sneakers for dance. They have a polyurethane split sole, which provides lots of flexibility, and they are quite comfortable. They work well for the most part for Latin and rhythm dances on hardwood, but they may be too sticky for smooth dances, where you need to slide your foot along the floor, and they also may be too sticky for linoleum floors for any dance.
Each person is different. It may take some experiments to find a good pair of dance shoes.