The main attraction of elliptic curve cryptosystems over other public-key cryptosystems is the fact that they are based on a different, hard problem. This may lead to smaller key sizes and better performance in certain public key operations for the same level of security.
Very roughly speaking, when this FAQ was published elliptic curve cryptosystems with a 160-bit key offer the same security of the RSA system and discrete logarithm based systems with a 1024-bit key. As a result, the length of the public key and private key is much shorter in elliptic curve cryptosystems. In terms of speed, however, it is quite difficult to give a quantitative comparison, partly because of the various optimization techniques one can apply to different systems. It is perhaps fair to say the following: Elliptic curve cryptosystems are faster than the corresponding discrete logarithm based systems. Elliptic curve cryptosystems are faster than the RSA system in signing and decryption, but slower in signature verification and encryption. For more detailed comparisons, see the survey article [RY97] by Robshaw and Yin.
With academic advances in attacking different hard mathematical problems both the security estimates for various key sizes in different systems and the performance comparisons between systems are likely to change.