What is the difference between turn rate and yaw rate in aircraft?
Rate of turn is easy to define: it is the rate at which the direction of aircraft motion changes with respect to the ground (rate of change of track). If the aircraft is not slipping and there is no change in wind, then it is equivalent to the rate of change of heading.

Yaw rate, on the other hand, is slightly messier. Technically, yaw rate of the rate of change of the z-axis Euler angle. As an industry standard, the Euler angle in the aviation industry is defined by the z-y-x rotation, which means that the yaw occurs first, followed by pitch, followed by bank. Therefore, yaw rate is technically exactly equal to the rate of change of heading.

In many applications and loose technical speaks, however, yaw rate may also refer to the third component of the angular velocity. Angular velocity is the instantaneous rate of rotation about its axis of rotation. If the aircraft is rotating about the vertical axis only, then the third component of the angular velocity is, again, exactly equal to the rate of change of heading. 

In a coordinated turn, however, the aircraft is pitched and banked, and therefore it is not rotating purely about the vertical axis. In this definition, then, yaw rate is not equal to turn rate.

As an aside, it's mathematically incorrect to call any component of the angular velocity a "rate", since angular velocity is not a derivative of any quantity, even though angular velocity can be transformed into Euler rates if you know the current Euler attitudes.