What is the difference between a common goldfish and a carp-like goldfish?

The carp-like goldfish is the koi, a beautiful (and huge) ancient carp variant that came from Asia somewhere.

The common goldfish are very commonly bred and sold as household pets, and have relatives such as comet goldfish.

They're from the same family and are equally good. Koi should not be kept in tanks unless they are very young. Common goldfish are great starting pets, living in a wide range of temperatures (comfortably 60-80 degrees f).

Basically, they are relatives in the goldfish family. The koi is often seen as a carp.