Turn Indicator - "Dog house" - Single needle pointing to reference points, indicates "rate of turn" only by precession around the longitudinal axis of the airplane (i.e. gyro is mounted in forre/aft design). (Edit : sorry gyro is mounted longitudinal giving information around the vertical axis.) You determine how fast you are turning as in saying 2 minute turn.
Turn Coordinator - "miniture airplane" points to "tick" marks. indicates "rate of turn" by using a 70% rate of yaw / 30% rate of bank combination. Gyro is mounted on an angle making it sensitive not only to yaw movements (nose right/nose left) but also to bank movements. (Information depicted is still "rate of turn" (combined).
In a coordinated turn, both indicators give proper information. In an uncoordinated turn, the TC is more "sensitive" and will indicate wing banking in the proper direction (where as TI needle will lag information).
And that's all I can say about that.