Difference between Inphase/Quadrature (I/Q) Mixer and I/Q Demodulator
A mixer is just a circuit that multiplies two signals.

The purpose of a mixer is to shift a signal in the frequency domain.

Normally a mixer multiplies a signal carrying information (some range of frequencies, e.q. received signal or signal that is going to be tranmsitted) with a signal without information, consisting of exact one frequency (from the local oscillator). 

An I/Q-Mixer consists of two ordinary mixers that each multiply the signal carring the information with the local oscillator signal, but one being 90° phase shifted.

A demodulator in general is a circuit that extracts the actual information of an RF signal.

There are many different demodulators (as there are many ways of modulation) and even for one particular modulation (e.g. FM, SSB) sometimes more than one type of demodulator is possible.

An I/Q demodulator is just some kind of demodulator using (among other subcircuits) an I/Q mixer as mentioned above.

Summary:
An I/Q mixer is a part of an I/Q demodulator.
An I/Q demodulator always contains an I/Q mixer, but I/Q mixers may also be used in circuits that are no demodulators (e.g. modulators).