The main differences are:

1. Different bracing patterns under the soundboard.
2. Flamenco guitars tend to be narrower.
3. Strings action is set much lower in flamenco models. This is done by lowering the bridge, nut and the saddle, and also by using a different angle to join the neck with the the body making the strings run more parallel to the neck and soundboard than they do in a typical classical guitar.
4. Because the action is lower, string buzzing is normal to a certain extent with a flamenco guitar. This would be totally unwanted with a classical model.
5. In a flamenco guitar Spanish solid cypress is traditionally used for back and sides and German spruce for the soundboard. In a classical guitar Indian rosewood is the wood of choice for back and sides and German spruce or cedar is used for the soundboard.
6. The Flamenco guitar has a large "tap plate" or "golpeador" to protect the soundboard from the characterisic percussive hitting (tapping???) of the soundboard with the player's nails

In recent years, the "flamenca Negra" (with its Indian rosewood back and sides) has become increasingly popular . This type of flamenco guitar has been adopted by a number of famous Spanish flamenco players. There is also the practical reason that that cypress wood has become more difficult to obtain and as a consequence more expensive.