xhuke has provided a good description but many instrument manufacturers tend to use the terms to define where the transducer data is processed to provide the measurement in the appropriate units. For example, the term transmitter can mean a device e.g a density meter, which transmits an output in the required units e.g. a 4-20mA signal proportional to density in kg/m3.
A transducer is then an instrument which transmits raw sensor data for remote processing. In the case of a vibrational density transducer this is a frequency. Frequency is proportional to density but must be corrected for a number of effects including temperature, velocity of sound, viscosity, pressure etc. This is then done in remote electronics.