The turn and bank instrument has the gyro mounted so that it reacts to yaw only and not to roll.
The turn co-ordinator has the gyro mounted at a 35 degree angle therefore it reacts to roll and yaw.
If you think about it when we turn an aeroplane we roll it into a banked attitude but this would not be seen on the turn and bank indicator until the aircraft starts to yaw around the turn.
In a nutshell think of the turn and bank indicator as a yaw instrument and the turn co-ordinator as a yaw and roll instrument.