Ballroom shoes are engineered wonders. First, the sole: as TruCelt says, they have to combine the ability to slip when you want with the ability to grip the floor at other times. The solution is suede (and a suede brush). Men’s shoes have suede on both soles and heels, women’s on just the soles (the heels are hard). In the US, most ballrooms won’t let you mess up their floors with rosin or baby powder or any other substance. Also, many insist on heel protectors on women’s shoes.
Second, the heel placement: other shoes can have the heel attached to the shoe anywhere from under the middle of your heel to in line with the back of your heel. Take the Melissa style. The heel is place directly under the middle of your heel, giving you the most stability. That’s very important if you are doing leaps, kicks, etc. You are landing with the full force of all your weight on 3 square inches- you want as much stability as you can get!
Third, the arch: Women’s shoes are made with a steel arch support. Men’s shoes usually aren’t because of the lower heel. Again, stability.
Fourth, cushioning: dance shoes should have cushioning all through the sole or your feet won’t even make it through an hour of dancing.
Fifth, straps: You want the shoe to stay on your foot!
There are usually different heel heights within a shoe style (although not offered by all distributers). You can get the Melissa style with a 2.5" through a 3.5" heel. The balance is different with each heel height and a good manufacturer will take that into consideration.
It used to be that you had to buy white satin and dye it to whatever shade you wanted. Now they offer nude or suntan shades, as well as black. (The theory is that nude/suntan is the best choice as they “disappear” on your leg, making it look longer and leaner. The fact is that, if you’re buying just one pair, nude goes with everything.)
Also, be careful to get a good fit.
The open toe (sandal) is generally for the Latin dances. The closed toed shoes are generally for ballroom styles. If you’re buying only one pair for both styles to begin with, most go with open toed shoes with a moderate heel (2- 2.5").