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Reading from "www.nadex.com" - the copy reads "Binaries are similar to traditional options but with one key difference: their final settlement value will be 0 or 100. This means your maximum risk and reward are always known and capped.".
Isn't that true when you are using traditional options? (Assuming the markets are the same.)
Addition: Can't you essentially replicate the payoff of a binary option using a vertical ATM option spread?
The text of your question doesn't actually match the question title. The answer to your title is of course yes binary options make sense. And as others have pointed out with binary options your reward is limited, and conversely the risk involved in writing them is less.
To answer your additional question you can replicate a binary option with a tight call spread around the strike (not ATM as you suggest). So for example, if you have a binary call struck at K, which pays off 1 if it's ITM and 0 if not, you can replicate that with $$ (C(K+\epsilon) - C(K))/\epsilon. $$ Where $\epsilon$ is typically the smallest strike unit you can trade.
Risk managing these positions can get tricky when the level of the forward gets close to the strike, as the greeks can change sign.
The key difference besides the cap is that there is nothing in between: its 100 or nothing (binary!) - with traditional options you have S-K as long as S>K (for the call).
You can find out more here: http://en.wikipedia.org/wiki/Binary_option
No. If you are long a vanilla option, your reward is unlimited. If you are short an option, your risk is unlimited.
They would make sense in certain narrow applications; one can perhaps think about scenarios where binary option might be the most efficient, quickest or easiest way to either benefit from a particular insight OR to hedge against some sort of event ... the real question is whether the volume in the markets for binary options will continue to sufficient to generate enough liquidity to render them as a practical alternative for different people who might have a reason to engage in either sides of the trade.
In other words, whether they make sense or not depends upon whether there are enough people who believe that other enough people will believe that they make sense. This is true for any any asset or derivative market. All markets fail (i.e. stop making sense) when people stop believing that other people believe that the market has stopped making sense for anyone.
For a long or short position in a vanilla put, the maximum risk/reward is known and capped. For a long position in a vanilla call, the maximum risk is also always known and capped. The maximum reward is therefore known and capped for a short vanilla call position. However, the reward is unbounded for a long vanilla call position, and therefore the risk is likewise unbounded for a short vanilla call.
Although binary option is not as liquid and common as European option, it makes perfect sense. There is only two possibility in payoff in your scenario: 100 or nothing. The cost of the option is therefore cheaper than an European option with everything else equal.
Binary option can be replicated by two call options around the strike (see diagram). Mathematically, the payoff can be defined by letting h infinitely small, exactly the same logic as @ldnquant has written.
