Sensor:
As the name suggests, it is a device which reacts to a physical, chemical or biological condition. It senses a change in physical, chemical or biological condition. It can be considered as a detector.
A Sensor can sense in any form usually electronic i.e. due to some mechanical change, it can react in electrical form. Thus there is a conversion, similar to that of a transducer.
A classic example is a thermocouple or a pressure sensor which might detect pressure and convert it into electric current (3-15 psi to 4-20 mA).
Transducer:
The conversion of energy from one form to another is known as Transduction. A Transducer serves for this purpose.
A Transducer is more than a sensor. It consists of a sensor / actuator along with signal conditioning circuits.
A signal conditioning circuit, by the name is a circuit which conditions the signal so that it is strong enough for further processing. A system might contain many stages before the signal finally reaches its destination to derive meaningful information.
So one way to differentiate Sensor and Transducer is that the output from a sensor may or may not be meaningful i.e. most of the times it needs to be conditioned and converted into various other forms. The transducer output is always meaningful.
The output of a motor is meaningful. The output of a loudspeaker is meaningful. They are transducers. A sensor is nothing but just a primary element which senses any physical phenomenon or it gives an indication in any change of the physical phenomenon.
We can say that every transducer is also (or has) a sensor but every sensor need not be a transducer. Sometimes it is. Sometimes in a sensor, there is no conversion at all. For example, thermometer, where the temperature is sensed and is directly measured. In a transducer there is always a conversion i.e. transduction. For example, RTD, Thermocouple etc where the temperature is sensed and the measurement is made in terms of voltage.
Thus we can say that a SENSOR may or may not have a conversion and it only senses. A TRANSDUCER always involves a conversion and also has signal conditioning involved.
Thank you!