There are elliptic curve variants of Diffie-Hellman, ElGamal, DSA and possibly other protocols/algorithms. I know that these elliptic curve variants have smaller key and ciphertext sizes which will make communications (or storage) more efficient.

Are there other advantages to the elliptic curve variants of these standard protocols/algorithms?

Specifically I am interested in speed and power consumption, but other potential advantages like resistance to side-channel attacks are also of interest.