This seems to be a favourite on the FIC test for some reason. I got asked it anyway.
The ball does the same on both so ignore that bit.
The turn indicator shows rate/direction of yaw only.
The turn co-ordinator shows rate/direction of yaw and also shows rate/direction of roll. This is due to the gyro being slightly inclined (30 degrees from memory) and thus sensing movement about the longitudinal axis.
Benefits of the turn co-ordinator are that it is more useful for limited panel flying and can also be coupled to a wing leveler.
Benefits of a turn indicator is that it will show you the correct direction of a spin even when inverted. Don't know how useful this is to most people though
FIS.